Israel Journal of Mathematics, Volume 2, Issue 4 (1964)

TWO TOPOLOGICAL PROPERTIES OF TOPOLOGICAL LINEAR SPACES* BY

CZESLAW BESSAGA AND VICTOR KLEE ABSTRACT

A topological and a geometrical-topological property, previously known only for normed linear spaces, are established here for much more general classes of topological linear spaces. Introduction. Throughout the present paper, E and E' will denote topological linear spaces (real scalars, separation axiom assumed). A convex body in E or E' is a convex set which has nonempty interior. Our two main results are as follows. THEOREM A. I f E is infinite-dimensional and admits a countable family of open (or closed) convex bodies whose intersection consists of a single point, then for each point p of E the spaces E and E ~ {p} are homeomorphic. ~ THEOREM B. Every closed convex body in E is homeomorphic with a closed halfspace or with the product of an n-cell by a closed linear subspace of finite deficiency n in E. t t These results were first established in I2] for Hilbert space, and were extended in 13] and [1] to arbitrary normed linear spaces, t Note that infinite-dimensionality is required for A but not for B. The topological property expressed in A has a number of interesting consequences; in particular, it implies that E admits a fixed-point-free homeomorphism of period two. Property B is useful in connection with the topological classification of closed convex bodies. Our methods are analogues or refinements of those employed previously, and as before the notions of gauge functional and characteristic cone will play an important role. When y is an interior point of a convex body U in E, the gauge functional of U with respect to y is the real-valued function/~uy defined as follows for all x e E:

pv,(x) = inf { 2 >O: -~(x - y)~ U } . Re~ived December 6, 1964. * This research was conducted at the University of Washington in 1963 when the first author was visiting there. The work of both authors was supported in part by the National Science Foundation, U. S. A. (NSF-GP-378). t, t t See the footnotes on the last page 211

212

CZESLAW BESSAGA AND VICTOR KLEE

[December

When y is the origin 0, we speak simply of the gauge functional of U and write /zv rather than/~u0. In several instances below, we shall define a transformation geometrically and leave to the reader the routine but occasionally tedious verification that the transformation is actually a homeomorphism. This can often be accomplished by expressing the transformation in terms of the appropriate gauge functionals and then making use of the well-known continuity of the function/tvy(x ) [ (x, y) e E x int U. For y e int U, we define cc, U = {x ~ u:/~uy(x) = 0}, ccU = ccyU - y, and csU = cc(U N (y - U)).

Thus the sets y + ccU and y + csU are the unions with {y} of, respectively, all rays and all lines which issue from y and lie in U. The sets ccU and csU, being independent of the choice of y ~int U, are called respectively the characteristic cone and the characteristic subspace of U. Note that the convex cone ccU is a linear subspace if and only if ccU = csU. NOTATION The interior, boundary, closure and convex hull of a set X are denoted by intX, aX, clX and con X respectively. The fact that X and Y are homeomorphic is indicated by X ~ Y. Set-theoretic addition and subtraction are indicated by U and ~ respectively, while + and - are reserved for algebraic operations. The real number field and the set of all positive integers are denoted by 9l and ~R respectively. Equality by definition is indicated by .= or = . . When x and y are distinct points of a linear space, the open segment connecting them is denoted by ] x, y [, the half-open segments by ] x, y] and ]'x, y [-, and the closed segment by J-x,y] The open and closed rays which issue from x and pass through y are denoted by ] x, y ( and [x, y ( respectively.

1. Three Propositions. The three propositions of this section are used later in proving Theorems A and B.

11

PROPOSITION.I:~L, II II) is an in:nite dimensional normed linear space,

then the linear space L admits norms I [ and I11 I II such that

I I= x +, and 0 < F(t) < 1, Vx- < t < x ÷ . In any case where x - > - ~ or x + < + ~ , we allow the possibility that search procedures may be finite and may have _ ~ for the last entry. Whenever we reach a point, we shall assume it has been searched, and thus we normalize F by assuming that F is continuous from the right in the positive half of the real axis and from the left in the negative half. The jump at 0, if there is one, is unimportant, as 0 is searched immediately at the outset. Thus, our answer will be the same whether we consider the given probability distribution or the

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ON THE LINEAR SEARCH PROBLEM

223

conditional distribution on the hypothesis t ~ 0. The distribution function for the conditional probability is continuous at 0, and thus it will not affect the generality of our result if we assume that F is continuous at 0. Suppose we choose a sequence x (") = {x[ ")} with X ( x (")) ~ too. A reasonable procedure would be to define Yi = lim,_~ o0x~") and prove that X(y) = m o , where Y = Yi. In fact, where possible, that is exactly our procedure. We must be wary, however of the possibility that x~")-~ 0, Vi, and this u n h a p p y circumstance can occur regardless of the nature of F, since for any search plan x, arbitrarily m a n y points can be added at the beginning arbitrarily close to 0, such that the change in X(x) is as small as we like. Furthermore, x t~ can diverge to + oo but not, as we shall see, if x - = - oo and x + = + oo. So let us take this case first. 2. LEMMA. I f X- = -- oO, X + = + o% then we can find a sequence (bi} such that Ix i [ < b, < ~ , k/i = 1, 2,-.-, holds for every search plan x with X(x) < 2m o. Proof. Let P1 = min(Pr(t < 0), Pr(t > 0)). Assume that xl > 0; the other case is dual. In this case,

21xil " Pl
2EIx21½-K • Dlx213 >

2E lx21- lx 13 = o Q.E.D.

4. LEMMA. I f F-(O) < o0, ~ > O, and x is any strong search plan, then we can :find a search plan y such that X(y) < X(x) + 5, Yl > O, and Y3 - Y4 >=K, where

K = K ( F ) is defined in the proof of the previous lemma. Proof. Perhaps x~ >= 0. I f not, let z = {z~} be chosen with 0 < z~ < x2 and I f z 1 is small enough, then X ( z ) < X ( x ) + 8 . If Xx>0, let z = x. I f z3 - z4 > K, we are through. I f not, then z ~1) = {z~~1)} defined by z} a)= Z~+z has X(z ~1)) =<X(z) < X(x) + e. Defining z (") by z~") = ",~(n-l)+2,we have X ( z c")) < X ( z t"-a)) whenever z(3"-1) -z(,,"-1) < K. Since z3(")= z3+2.~must eventually exceed K (Recall that F ( K ) - F(O) < ½Pr(t > 0).), we must come to a first n for which ,-a"~")- ,-4"t")=> K. Then if we set y = zC",)we have X(y) < X(x) + e. Q.E.D.

zl=x~-l, Vi>2.

5. LEMMA. I f X ( x ) < 2 m o , x x > 0 , x 3 - x 4 > K, and x- < a < O < b < x +, then x I e [a, b] for only no values of i at most, where n o depends only on F, a, b, and K. Proof. Let P = rain {Pr(x- < t < a), Pr(b < t < x + )}. Then if, for example, a __<x2, =< 0, we see immediately that


nx, we have

fy

'lk--1 X(y,t)dF(t)- f ' ~ - - ' X ( w C"),t)dF(t) 2k

max{no, ns}, we have

f,

':~-' X(y, t)dF(O
0, z~* ) - z(4")> K, and so that X(z (~)) ~ mo as n ~ oo. I f {z~~)} is b o u n d e d for each i as n ~ ~ , then the p r o o f given for T h e o r e m 6 will apply here. In the other case, let k be the least value of i for which {z}~)} is unbounded as n ~ oo. Choose a subsequence {z (~J)} of {z (~) } such that {z}*J)} converges for each 1 < i < k, while {z~~j)} ~ + ~ , and such that X ( z (~)) 0 ; the other case is dual. Choose any Yx with x2 < Yl < 0 and y ~ i ( F ( y t ) - F(0)) > (1/xi), so that F(yx) - F(O) < (yl/xO, and define y~ = x~_ 1, V i => 2. Then X ( y ) - X ( x ) = 2 [ y l ( f ( 0 ) - F(yl) - 1) + x l ( F ( y l ) - F(0)] < 2

yl(F(O)- F(yl)-

1) + x 1 •

= 2yj(F(0) - F(yO) < O, so that X(y) < X(x).

Q.E.D.

228

ANATOLE BECK

Proof of Theorem. 1. Direct consequence of "l?heorems 6, 8, 11, and 13 and Corollaries 7, 9, 10 and 12. Q.E.D. In fact, Theorem 12 shows a little more than promised. It shows that if /v-(0) = oo (resp. F + ( 0 ) = oo), then there is no minimal search procedure with x I > 0 (resp. xl < 0). Thus, if F - ( 0 ) = 0% F+(0) < ~ (resp. the reverse), then we see that there is a minimal search procedure, x, and xt < 0 (resp. x 1 > 0). Thus, in this limited case, we have an indication of the direction of the first entry.

REFERENCE 1. Franck Wallace, On the optimal search problem, Technical Report No. 44, University of New Mexico, October, 1963. THE HEBREW UNIVERSITY OF JERUSALEM, AND THE UNIVERSITY OF WISCONSIN

S H A D O W SYSTEMS OF CONVEX SETS BY G. C. SHEPHARD

ABSTRACT

An s-system of convex sets is the system of shadows of a given convex set cast on to a subspace by a beam of light whose direction varies. Here the convexity properties of s-systems are investigated, and, in the final section, a relationship with the projection functions of convex sets is established. In three-dimensional space, the shadow of a convex set cast on to a plane by a parallel beam of light is a convex region. If we let the direction of the beam vary, we get a system of convex regions in the plane which will be called a shadow system of convex sets, or, more briefly, an s-system. The purpose of this short paper is to investigate the properties of s-systems. In particular it will be shown that if an s-system is parametrised in a suitable way, many geometrical functionals such as the volume, surface area, diameter, etc., are convex functions of the system parameter. Although there is no exact theory of duality in the study of convex bodies, s-systems seem, in some sense, to play the part of duals to Minkowski concave systems. This duality arises because, whereas an s-system consists of the projections of some convex set in higher-dimensional space, a Minkowski concave system may be considered as arising from the parallel sections of a such a set [2, p. 33]. S-systems are closely related to the process of Steiner symmetrisation [-2, p. 69], and, in a one-dimensional form, occur implicitly in the works of many authors (compare, for example, the continuous symmetrisation of P61ya and Szeg6 ['5, p. 200] and the linear parameter system of Rogers and Shephard [-6, p. 95]). §1. Definitions and Elementary Properties. In Euclidean space o f n + 1 dimensions, E n+ l, let ~ be any non-zero vector, K be any closed bounded convex set, and . ~ any hyperplane (subspace of n dimensions). Then we define S(~, K, . ~ ) (the shadow of K on ~ in the direction ~) to be .g~nZ(K, ~) Received December 30, 1964. 229

230

G.C. SHEPHARD

[December

where Z(K, 0 is the cylinder {x + t~ I x e K, - oo < t < oo} containing K with generators in the direction ~. Let a be any fixed vector in E n+1 not parallel to a~f. Since the definition is affm¢ invariant there will be no loss of generality in assuming that a is a unit vector normal to ~,~. Let u be a variable vector parallel to ~ , and let K(u) = S(a + u, K, ~ ) . Then the system of convex sets {K(u)}, as u varies, is called an s-system, and u is the system parameter. The s-system will be said to originate from the set K cE n+l

•

Since, dearly, S(a + u, x + (a, ~ ) = x - (u for any real number (, an alternative definition of K(u) is K(u) = {x - ~u Ix + ~a ¢ K} for each u. Written in this way, it is easy to see that if u is restricted to lie on a line, we obtain the linear parameter system of [6]. Since S(~,K, ~ ) is an affine image of the orthogonal projection of K on to a hyperplane normal to ~, many elementary properties of s-systems follow immediately from the corresponding properties of orthogonal projections. For example: I f all the sets K(u) of an s-system are centrally symmetric, then so is the set K from which they originate. The corresponding result for orthogonal projections was first proved by Blaschke and Hessenberg, see [2, p. 124]. As a second example, we mention the (rather surprising) fact if {Ki(u)} and {K2(u)) are two s-systems such that (2)

vn(Kl(u)) < vn(K2(u))

for all u, then it is possible for (3)

v~+I(KI) > vn+l(K2).

(Here vn(X) means the n-dimensional volume or content of the set X.) The corresponding result for orthogonal projections must have been known for a long time, but does not seem to appear in the literature; we therefore give an example below. The question of whether (2) implies vn+ ~(K1) < Vn+l(K2) if we restrict K1 and K2 to be centrally symmetric is still open. An answer would be interesting since this question is dual (in some sense) to an unsolved problem of Busemann and Petty [3, p. 88] about the cross-sections of centrally symmetric bodies. Let K1 be any ball in E s, and K~ be any non-spherical body of constant brightness ([2, p. 140] and [1, p. 151]) whose orthogonal projection (and therefore shadow) in any direction is equal in volume to that of KI. By Cauchy's surface area formula [2, p. 48] and the isoperimetrictheorem [2, p. 111] it follows that vn+l(Kt) > vn+I(K~). If we dilate K~ slightly, we obtain

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231

a body K 2 which satisfies (2), and, if the dilation is small enough, will satisfy (3) also. Let Ho, Hi be two given convex sets in ~ . Then it will not, in general, be possible to find an s-system which contains them both. A criterion for this is: (4) Let Ho, H 1 be any two closed, bounded convex sets and u o, u I be any two vectors in E n. Then there exists an s-system (K(u)) such that K(uo) = Ho, K(Ul) = H1 if any only if P(Ho, ~ ) = P(H1, ~), where P(H, ~ ) is the orthogonal projection of H on to a hyperplane ~ c ~v normal to the vector Uo - ul. The corresponding criterion for more than two sets H~ is not known. Clearly (4) is necessary since, by the properties of orthogonal projection, both P(K (Uo), ~ ) and P(K(ui), ~1) are equal to P(K, ~). It is also sufficient, for if a is any unit vector in E n+ 1 normal to the hyperplane ~ ' in which Ho, H1 lie, then we may put

K = Z(Ho,a + Uo) n Z ( H I , a + ui) and it is easily verified that Ho = K(uo) and H1 = K(Ul). This proves (4). There will be many s-systems containing the given sets Ho and H i , but for any such system {K'(u)}, it is clear that K' c K . Hence the system defined above is, in an obvious sense, the 'maximal' one. It may be called a linear s-system by analogy with a linear Minkowski system, which is the 'minimal' concave system containing two given sets. If rio - K(uo), H1 = K(u~), then the system of sets K((1 -O)uo+Ou~)(O 2, v,(K(ul, ..., ut)) is not in general, a convex function of T jfor t > 1, but is, in certain special cases. For example it is so if K is a simplex of at most n + 2 dimensions I-7, p. 307] or if K is a vector sum of line segments [4, p. 20]. The fact that v,(K(ul ,..., uf)) is, for all K, a convex function of each parameter u s separately corresponds to the fact that/~(K, T ±) is a weakly convex function [4, p. 34], i.e. is convex on the generators of G~ +t. REFERENCES 1. W. Blaschke, Kreis und Kugel. Viet, Leipzig 1916. Reprint: Chelsea, New York, 1948. 2. T. Bonnesen and W. Fenchel, Theorie der konvexen K6rper. Springer, Berlin, 1934. Reprint: Chelsea, New York, 1948. 3. H. Busemann and C. M. Petty, Problems on convex bodies. Math. Scand., 4 (1950, 88-94. 4. H. Busemann, (3. Ewald and G. C. Shephard, Convex bodies and convexity on Grassmann cones, Parts I-IV. Math. Annalen, 151 (1963), 1--41. 5. G. P61ya and G. Szeg~, [soperimetric Inequalities in Mathematical Physics, Princeton University Press, 1951. 6. C. A. Rogers and G. C. Shephard, Some extremalproblemsfor convex bodies. Mathematika, 5 (1958), 93-102. 7. G. C. Shephard, Convex bodies and convexity on Grassmann cones, Part VI. Jour. London Math. Soc., 39 (1964), 307-319. UNIVERSITY OF BIRMINGHAM, BIRMINGHAM,ENGLAND

NONNEGATIVE MATRICES WITH STOCHASTIC POWERS* BY

DAVID L O N D O N ABSTRACT

Matrices with nonnegative elements, which are nonstochastic but have stochastic powers, are considered. These matrices are characterized in the irreducible case and in the symmetric one. 1. Introduction. In this paper we consider square matrices with nonnegative elements which themselves are not stochastic, but for which a certain power is stochastic. In §2 we deal with nonnegative irreducible matrices, and in §3 with nonnegative symmetric matrices. In each of these cases we obtain a characterization of the nonstochastic matrices of the corresponding class which have stochastic powers. Our characterizations are constructive and enable us to build effectively the corresponding matrices. A very special case of our second result, the characterization of all 3 × 3 nonnegative symmetric matrices A which are nonstochastic, but for which A 2 is stochastic, was obtained earlier as a byproduct of the proof of a certain matrix inequality [2, Remark 3 following Theorem 1]. The main tool used in this paper is the Perron-Frobenius theorem [1, p. 53]. Let A = (aij) be a n × n nonnegative irreducible matrix. By the Perron-Frobenius theorem, A has a dominant simple positive characteristic value ~ = ~(A). If at ,-'-, c~h= a are all the characteristic values of A with modulus ce, then (~k ~(.0k k = 1,...,h, where co = e 2~i/h. If h = 1 A is primitive. If h > 1 A is cyclic o f index h. If A is cyclic of index h, then there exists a permutation matrix P such that =

m

0 0

(1.1)

pAp r =

D

A1 0 0 A2 0 \ \ \

0 0

\ \ 0 0 Ah 0

0

Ah_ 1 0

Received December 15, 1964. * This paper represents part of a thesis submitted to the Senate oftheTechnion-Israel Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Science.The author wishes to thank ProfessorB. Schwarz for his guidance in the preparation of this paper.

237

238

DAVID LONDON

[December

The null matrices in the main diagonal are squares of orders nk, k = 1, ..., h. (1.1) is the Frobenius normal form of A. Let rl, ..., rh be the characteristic values of PAP T corresponding respectively to ~1, "", ~h. rh is positive (r h > 0). Write rh = z 1 4 - . . . 4 - z h , where Zk is a vector of order nk, and the symbol 4- indicates direct sum. (If u = (ul,...um) and v = (vl,-..,v,), then u 4- v = ( u l , ' " , u m , v l ' . . , v , , ) ) . We have r k = z t 4- (.okz2 4- (o2kz3 4- "" 4- oj(h-1)kzh,

k = 1,...,h.

We end this introduction by a definition. Let B = (b/y) be a nonnegative m × n matrix. If

bij = fl,

i = 1,..., m,

j=l

then B is fl stochastic or generalized stochastic. If fl = 1 then B is stochastic. We remark that usually this definition is given only for square matrices. However, for our purpose it is convenient to use it for rectangular matrices. 2. Nonnegative irreducible matrices. Let A be a nonnegative irreducible matrix which is not stochastic. In this section we obtain a necessary and sufficient condition for some power of A to be stochastic. THEOREM 1. Let A be a nonnegative irreducible square matrix and let m > 1 be a positive integer. Let H be the cyclic permutation H=

(12,... h),

and let (2.1)

H m = CIC2". C,

be the representation of H mas the product of disjoint cycles. A is not a stochastic matrix while A m is stochastic if and only if (I) A is cyclic of index h, where (h, m) > 1. (II) There exist positive numbers fli, i = 1, ...,h, such that the matrices A i appearing in the Frobenius normal form (1.1) of A are respectively fldflt+l stochastic.(*) The numbers fl~fulfill the following two conditions: (A) They are not all equal. (B) Every two numbers with indices belonging to the same cycle in (2.1) are equal. Proof. First we prove that the conditions (I) and (II) are necessary. Let A be a nonnegative irreducible matrix which is not stochastic while A = is stochastic. * Here and in the sequel the indices are taken modulo h.

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NONNEGATIVE MATRICES WITH STOCHASTIC POWERS

239

As A s is stochastic, 1 is the dominant characteristic value of A s and e=(1, ..., 1) is a corresponding characteristic vecor. Returning to A, it follows that 1 is the dominant characteristic value of A. As A is not stochastic, e is not a characteristic vector of A. Assume A is primitive. Then 1 is a simple characteristic value of A m and the only characteristic vector of A s corresponding to the characteristic value 1 is the characteristic vector of A corresponding to 1. But as this vector is different from e, it follows that A cannot be primitive. Hence, A is cyclic and can be represented by the Frobenius normal form (1.1). As P A P r is only a cogredient permutation of the rows and columus of A, we may change in the above considerations A and A m respectively with P A P r and P A m P r. Let cq ,..., eh = 1 be all the characteristic values of P A P r(or of A) with modulus 1, and let r l , . . . , rh be the corresponding characteristic vectors. As quoted in §1 we have (2.2) (2.3)

c~k = cok, co = e z~i/h,

k = 1,..., h,

r k ~--- Z 1 4- (DkZ2 4- (D2kZ3 4- "'" 4- c o ( h - l ) k z h ,

k = 1,-..,h.

As e is a characteristic vector of P A " P r c o r r e s p o n d i n g to 1 while it is not a characteristic vector of P A P r, there exist integers kt , ..., kt ," 1 =< kx < k2 "" < kz < h, l > 1, such that (2.4)

co,,~, = o)mk2. . . . .

comk,= 1,

and also numbers d r , "-, d~ such that (2.5)

dxrkl + d2rk~ + "" + dlrkz = e.

(2.3) and (2.5) imply (2.6)

Zl(dx

+ "'" nt-

dr) 4- "'" 4- Zh(dlco

(h-1)kt

+ ... + d , c o ( h - l ) k , ) =

e.

Let ei = (1, ..., 1), i = 1,..., h, be a vector of order nt. From (2.3), (2.6) and the fact that rh > 0 it follows that there exist positive numbers/~a, "",/3h such that (2.7)

rh = flxel 4- f12e2 4 - ' " -Jr flheh.

As rh is a chracteristic vector of P A P r corresponding to the characteristic value 1, we obtain from (1.1) and (2.7) p A p r r h = B2Ale2 4- "'" 4- PhAh-teh 4- ~lAhel = ~1el 4- "" 4- flhen.

Hence, (2.8)

Aiei+t = ~-ff-~-~ e~,

i = 1,...,h.

Pi+I

From (2.8) follows that A i is a//i//~+l stochastic matrix.

240

DAVID LONDON

[December

We have now to show that fit fulfill the conditions (A) and (B). PAPris not stochastic and therefore not all the matrices At are stochastic. As At is fl~/fl,+ stochastic, it follows that not all the numbers fl~ are equal. (A) is thus proved. To prove (B) denote the blocks of PAP r in the partitioning (t.1) by Aii, i,j = 1,..., h, and the blocks of pAmp r in the same partitioning by" Llt,A(m)j.We have AiS = { At'

(2.9)

0 ,

j = i + 1 (mod h) j~:i+l

(modh)

and h

(2.10)

A}~") =

~

Aik, ak~k2 ..... Ak,.-,, i"

kt,-..,km- t =1

From (2.9) and (2.10) follows (2.11)

A!m')={~ tAi+l"''''At+m-l',s ,

j=-i+m

(modh)

j~i+m

(modh).

As pAmp r is stochastic, all the matrices A t,(m)i+m are stochastic. As At is fit~fit+ 1 stochastic, it follows from (2.11) that Atl~)+m is fli

fli+X

fli+m-1 __ fit

stochastic.

Hence,

(2.12)

fit = fli+~,

i = 1, ...,h.

The permutation H ~ carries i into i + m and therefore i and i + m belong to the same cycle in (2.1), and so (2.12) is equivalent to (B). (B) is thus proved, and the proof of (II) is established. We have already proved that A is cyclic of index h. To complete the proof of (I), we have to show that (h, m) > 1. This fact follows easily from (2.2) and (2.4). The proof of the necessity part of the theorem is completed. We now prove that the conditions (I) and (II) are sufficient. Let A be a matrix which fulfills the conditions (I) and (II). From (A) follows that PAP r, and therefore A too, is not stochastic. According to (2.11) the matrix A~."~+mis flt/flt+~ stochastic. (B) is equivalent to fit = fit+m, and so ~(m) "*t,t+m is stochastic. Hence, PA~P r, and therefore A mis stochastic. The proof of Theorem I is thus completed. REMARK 1. We have H h = (1)(2)... (h), and so for m = h the condition (B) holds for any fit. In this case it is thus sufficient that the condition (A) holds. From this we conclude that if A is a nonstochastic cyclic matrix of index h and if A m is stochastic, then A s is also stochastic and so any power A rex, where ml = m (mod h).

1964]

NONNEGATIVE MATRICES WITH STOCHASTIC POWERS

241

REMARK 2. Let m and h be positive integers (m, h ) > 1. By the sufficient conditions of Theorem 1 we can construct all the matrices A which are nonstochastic and cyclic of order h, and for which A '~ is stochastic. As ( m , h ) > 1, there is more than one cycle in the representation (2.1), and so we can find positive numbers fl~, i = 1, ...,h, for which both (A) and (13) hold. Let Ai, i = 1,--.,h, be fli/fli+ 1 stochastic matrices chosen so that their dimensions fit the structure of (1.1) and so that A is cyclic of index h. Using the Aj's, we construct A according to (1.1). 3. Nonnegative symmetric matrices. Let A be a nonnegative symmetric matrix which is not stochastic. In this section we obtain a necessary and sufficient condition for some powers of A to be stochastic. Let us first define a class of matrices 9~. A matrix A belongs to the class 9~ if and only if A is a n × n nonnegative symmetric matrix, A is not stochastic and there exists a natural number m for which A s is stochastic. Let A eg~n . A s is thus stochastic while A is not stochastic. It is necessary that the multiplicity of the dominant characteristic value of A m is greater than the multiplicity of the dominant characteristic value of A. Hence, m is even. As the multiplicity of the dominant characteristic value of A m is equal for all the even re's, it follows that i f A E 9.I,, then A s is stochastic if and only if m is even. In the following theorem we characterize the classes 9~, by a recursive procedure. The structure of the class 9~ is determined by the structure of the classes 9.Ira, m < n. As the class 9~1 is void, we can by this procedure determine the structure of 9~ for any n. THEOREM 2. Let A be a n × n matrix. (1) I f A is reducible, then Ae9~, if and only if there exists a permutation matrix P such that

Bn - k

k is an integer for which the inequality (3.2)

n

-- < k < n 2 =

holds. B k and Bn_ k are respectively k × k and (n - k ) × ( n - k) matrices, and at least one of the following two conditions (3.3)

holds. I f only one of these conditions holds, then the matrix for which the condition does not hold is symmetric and stochastic.

242

DAVID LONDON

[December

(2) I f A is irreducible, then Aeg.l, if and only if there exists a permutation matrix P such that (3.4)

A = pr

p.

A~

0

0 indicates square null matrices, k is an integer for which the inequality n

(3.5)

-~- < k < n

holds. A1 is a k × ( n - k) matrix [ ( n - k)/k] 1/2 stochastic and its transposed AT is [k/(n - k)] 1'2 stochastic. Proof of (1). First we prove the necessity part. Let A be a reducible matrix belonging to 9~,. As A is reducible and symmetric, there exists a permutation matrix P for which (3.1) holds, where Bk and B,_ k are symmetric matrices. It is obvious that P can be chosen so that (3.2) holds. We have

]

0 A 2 = pT

2 nn-k

p.

As Aeg~,, A 2 is stochastic and therefore Bk2 and Bn-k 2 are both stochastic. As A is nonstochastic, at least one of the two matrices Bk and Bn_ k is nonstochastic. For the matrix which is nonstochastic the corresponding condition in (3.3) holds. If the other matrix is also nonstochastic, then (3.3) holds for this matrix too. If the other matrix is stochastic, then it is symmetric and stochastic. It is easy to verify that the conditions are also sufficient. Proof of (2). Let us begin with the necessity part. Let A be an irreducible matrix belonging to 92[,. According to Theorem 1, A is cyclic. As A is symmetric, it is cyclic of index 2 and so there exists a permutation matrix P for which (3.4) holds. It is obvious that P can be chosen so that (3.2) holds. According to Theorem 1 there exist positive numbers fll and f12, fll v~ f12, such that AI is fll/fl2 stochastic and AT is fl2/fll stochastic. As A 1 is a k x (n - k) matrix, we obtain

kfl2

fl-11"

Hence,

A1 is thus [(n - k)/k] 1/2 stochastic and AIr is [k/(n - k)]l/2stochastic. As fll # f12, it follows that the sign of equality in the lefthand side of (3.2) does not hold, and so (3.5) holds.

1964]

NONNEGATIVE MATRICES WITH STOCHASTIC POWERS

243

The sufficiency part follows by direct computation of PA 2pr. The proof o f Theorem 2 is completed. We shall now discuss the structure of the classes 9~n for n up to 4. n"l.

As already mentioned 9.It is void. n

m.~ 2 ,

(1) A reducible. (3.2) implies k = 1, n - k = 1. As 9~t is void, the condition (3.3) cannot be fulfilled, and so there are no reducible matrices in 9~2. (2) A irreducible. No natural k exists for which (3.5) holds, and so there are no irreducible matrices in 9.I2. Conclusion: 9.12 is void. n=3. (1) A reducible. (3.2) implies k = 2, n - k = 1. As the classes 921 and 9.I2 are void, the condition (3.3) can not be fulfilled, and therefore there are no reducible matrices in 993. (2) A irreducible. (3.5) implies k = 2, n - k = 1. At is a 2 × 1, l/x/2 stochastic matrix, and A has the following form m

m

1

0

(3.6)

A = pr

1

0

- -

1

1

p.

0

,/5

m

There exists 3 distinct matrices of the the form (3.6). Conclusion: 9~3 includes precisely the three matrices given by (3.6). From this conclusion follows the result mentioned in the introduction. n--4. (1) A reducible. (3.2) implies k = 2 ; 3 and so n - k = 2 ; 1 respectively. As 9.I2 is void, there remains only the possibility k = 3, n - k = 1. Let B3 be one of the three matrices belonging to 993. A has the following form 0 (3.7)

B3

A = pr

0

0 0

0

0

1

P.

244

DAVID LONDON

There are 12 distinct matrices of the form (3.7). (2) A irreducible. (3.5) implies k = 3, n - k = 1. At is a 3 × 1 matrix, stochastic, and A has the following form

11fg

B

(3.8)

1

0

0

0

~-~

0

0

0

~-~

0

0

0

~_

1

A = pr

1

1

1

P.

43 1

There are 4 distinct matrices of the form (3.8). Conclusion: 9I, includes 12 reducible matrices given by (3.7) and 4 irreducible matrices given by (3.8). REFERENCES 1. F. R.. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New-York, 1959. 2. D. London, Two inequalities in nonnegative symmetric matrices, (to appear in the Pacific J. Math.) TECHNION-ISRAELINSTITUTEOF TECHNOLOGY, HAIFA, ISRAEL

ON PROJECTIONS AND SIMULTANEOUS EXTENSIONS* BY

DAN AMIR ABSTRACT

Let Xb X2 be subspaces of a completely regular space X. The bounded linear extension of C(X1 0 )(2) into C(X) are related to the projections of norm < 3 from C(X0 + C(X2)onto C(X). 1. Introduction. C(X) denotes the Banach space of all bounded continuous real-valued functions on a topological space X, with the supremum norm. If B is a subspace of X, a simultaneous extension is a linear operator E from C(B) to C(X), such that for each f in C(B), Ef is an extension o f f . If R denotes the restriction operator of C(X) to C(B), then a simultaneous extension is a linear right inverse of R. When a bounded simultaneous extension exists, C(B) is isomorphic to the subspace EC(B) of C(X), and P = ER is a projection (all "projections" in this paper are linear and bounded) of C(X) onto this subspace. If X is metric and B is closed in X, then there exists a simultaneous extension E of C(B) to C(X) with norm 1 [2]. In the general case a bounded simultaneous extension may fail to exist: Let X = fin (the Stone-Cech compactification of the discrete sequence N), and B = f i N - N. As proved in [1], C ( f l N - N) is not isomorphic to a direct factor of C(flN). Corson and Lindenstrauss [4], found recently, for every k > 1, a pair B c X of compact Hausdorff spaces, such that there is a simultaneous extension of C(B) to C(X) with norm k, but no one with smaller norm. Another simple relation between projections and simultaneous ,'extensions was observed by Dean [3]: Let Co(X, B) denote the subspace of C(X) of functions vanishing on B. If E is a bounded simultaneous extension of C(B) to C(X), then I - ER is a projection of C(X) onto Co(X, B). If R has a bounded (not necessarily linear) right inverse Q on C(B) (e.g. when B is closed and X is normal--by Tietze's theorem), the converse is also true: If P is such a projection, define E = Q - PQ. E does not depend on the choice of Q (if Q' is another right inverse of R, then (Q - PQ) - (Q' - PQ') = (Q - Q') - P(Q - Q') = 0), and is a bounded linear extension. In this paper we study a less immediate relation between projections and simultaneous extensions: Suppose we have two pairs: B1 c X1, B2 c X2 and a homeoReceived January 28, 1965. * This work was supported in part by National Science Foundation grant NSF GP-2026. 245

246

DAN AMIR

[December

morphism h of BI onto B 2 . We can "paste" the spaces X1 and X2 along the Bi by identifying all the points s in B 1 with the corresponding hs in B 2 , the quotient space X having the quotient topology. C(X) is naturally identified as a subspace of C(X1) ~ C(X2). The theorem relates projections of C(XI)O)C(X2) onto C(X) to simultaneous extensions from C(B) to C(X), where B is the image of the Bj under the quotient map. This is done for the case where the Bi are closed and nowhere dense, the Xr--completely regular, and there exist norm preserving extensions Qi (not necessarily linear) of C(Bj) to C(Xi) (RjQj is the identity on C,(B,)) (i = 1, 2). 2. Tn~OREM. (In the conditions

ofC(B) toC(X), with

specified above) if P is a projection of

IIEII 0 be arbitrary. In the open set U = {s e X 1 - B;[ Qif(s)- f (x) ]< ~} choose a point t that satisfies also: [P(Qlf~)O)(t) - P ( Q l f ~ 0)(x) [ < e. Take a function g in C(X) such that 0 < g < 1, g(t) = 1 and g vanishes out of U. Consider the function F = ( - Q l f ~ Q 2 f ) + [1 + f ( x ) ] g which belongs to C(Xx) ~ C(X2) and satisfies IF] < 1 +

• (PF)(t) = ( Q l f ~ Q2f)(t) - 2P(Q~f@ 0)(t) + [1 + f ( x ) ] g(t) = 1 + 2f(x) - 2w~f(x) + [Q~f(t) - f ( x ) ] + 2 [P(Q~f~) 0)(t) - P(Q~f@ 0) (x)] by the choice of t, we have: (1 + e)II P II > (PF)(t) > 1 + 2f(x) - 3~ - 2W~f(x), and as 8 was arbitrary, we can conclude (by symmetry) that (2)

Wlf(x)~_f(x) - ½ ( I I P i l -

1) for all x e B ; f ~ C ( B ) with Ifl < 1 and i = 1,2.

Combining (1) and (2), we get also the upper bounds: WJ(x) < ½( ]1P I1 - 1). A very similar method gives us the same upper bound for x e X i - B: Take gEC(S) such that 0__< g=< 1, g(x)= 1 and g vanishes out of {s ~ XI - B; [Q~f(s) - Qlf(x) l < e}. Consider the function F = ( - Q~f ~) Qzf) + [1 +f(x)]g. IF l < 1 + e and (PF)(x) = - (Qlff~Q2f)(x) + 2Wlf(x) +

1964]

ON PROJECTIONS AND SIMULTANEOUS EXTENSIONS

+ 2g(x) = 1 + 2w, f(x), hence ( - f ) instead off, we get: (3) --

247

Wlf(x) __ sup {If(x) I ;x~B}-½ (11PII- 1) P 11)(by (2)), combining these results we get:

(4) o< ~(3-11PII)0 for a sequence n of positive upper density then lira sup g(n) = cx~. Let a , < a2 < "" be an infinite sequence of integers and denote by f ( n ) the number of solutions of n = a~ + aj. An old conjecture of T u r i n and myself states that if f ( n ) > 0 for all n > no then lim sup, = ~ f ( n ) = m. A stronger conjecture (which nevertheless might be easier to attack) states that if at < ck z then limsup,=~of(n) = m. Both these conjectures seem rather deep. I could only prove that ak < ck z implies that the sums at + aj can not all be different [6] (c, c t , c2,.., denote absolute constants). In view o f the difficulty of these conjectures it is perhaps surprising that the multiplicative analogues of these conjectures though definitely non-trivial are not too hard to settle. In fact I shall prove the following.

THEOREM 1. Let b~ < b 2 < ... be an infinite sequence o f integers. Denote by g(n) the n u m b e r of solutions of n = bib j . Then

(1)

g(n) > 0 for all n > no

implies

(2)

limsup g(n) = m n=OD

Define

B(x) = E 1 bt_~x

A well known theorem of Raikov [5"] states that (1) implies that for infinitely many x

(3)

B(x) > clx/(log x) ~/2

Thus to prove Theorem 1 it will suffice to show that if (3) holds for infinitely m a n y x then (2) follows. In fact I shall prove stronger results. Denote by ut(n ) the smallest integer so that if b: < --. < bt < n, t = u~(n) is any Received December 17, 1964. 251

252

P. ERD~)S

[December

sequence of integers then for some m, g(m) > I. Theorem 1 would follow from ul(n) = 0(n/(log n)t/2). THEOREM 2. U2k(n) < e2 l o ~ n log log n) k+l In a previous paper I [1] proved that (4)

II(n) + c3nal4/(logn) 3/2 < u2(n) < H(n) + c4 ha~4.

II(n) denotes the number of primes not exceeding n and IIk(n ) denotes the number of integers m > n the number of distinct prime factors of which does not exceed k. The right side of (4) can in fact be stregthened to (5)

u2(n ) < II(n) + csna/4/(log n) 3/2

I do not prove (5) in this paper. (4) and (5) suggest the possibility of obtaining an asymptotic formula with an error term for ut(n) also for l > 2. I am going to outline the proof of THEOREM 3. Let 2 k- 1 < l < 2 k. Then nilog log n) k- t ul(n ) = (1 + o(1) (k - 1)l logn " Finally I am going to prove the following THEOREM 4. To every c and l there is an n o = no(C, l) so that if n > no and b I < ... < b s < n is such that the number N(n) of integers t < n which can be written in the f o r m bib j is greater than en then there is an m with g(m) > I. Theorem 4 clearly implies Theorem 1, but not Theorems 2 and 3. Our main tool will be the following LEMMA. Let $ 1 , . . . , Sr be r sets of integers, S~ has N t elements ( N 1 > ... > N,) x (0, I < j < N , . Let u s < u 2 < . . . < u t be a sequence of integers where each uj, 1 ~ j < t is of the formI-[.~=l xt°(i.e, every u can be written as the product of r integers one f r o m each Si). Then if (6)

3': t> ~ 1-IN, Nlr 2r-I i=1

there is an m so that the number of solutions of m = UjIUj2

IS at least 2"-2. To each integer of S~, 1 < i < r we make correspond a vertex and to each

1964]

MULTIPLICATIVE REPRESENTATION OF I N T E G E R S

253

uj = l-i:= 1 x~°, we make correspond the r-tuple { xji}! 0 1 x 2

> ... >

/=1

where the x}~) and x}2~are the p's and q's and xrig x, =~ x(l)and , I-Ik= 1 x} 2) satisfy (27). x~~) and x~z) we will call the i-th coordinate of v~k]respectively .~j~.~k)Clearly p~ and q~ must be the first coordinates of any possible solution of (29). To see this observe that (27) implies k

I-I piqi < n a/2 < n/log n i=2

and hence (27) can be satisfied only if the first coordinates are Px and q t . Assume that the first i - 1 coordinates of a solution vii • (k)^~(29) has already been chosen. ut I claim that there are only two possible choices for the i-th coordinate o f "u j l(k)• To show this it will suffice to prove that only one p and only one q can possibly occur as the i-th coordinate of U" j(k)If this is not so we assume that both l • It t t vj=xl...x~_lpux~+t...xk and v j = x l . . . x i _ l p v x i + l . . . x k would be solution of (29). But then clearly (32)

vjr > =Xl...Xi_lPu,

tl

O j < X l ' . . X i _ I p k.

Hence by (27) and (32) , ,, k pul/2 > log n > vy/vj > Pu/Pv > log n

an evident contradiction.

1964]

MULTIPLICATIVE REPRESENTATION OF INTEGERS

257

The fact that the first coordinates of every solution of (29) must be p~ and ql and the fact that for i > 1 there are most two choices for the i-th coordinate of v(k) h immediately implies that (29) has at most 2 k- 1 solutions. Thus by (28), (26) is proved. To complete the proof of Theorem 3 we have to show (33)

u2~(n) < (1 + o(1))

n(log log n) k- i (k - 1) ! log n

To prove (33) it suffices to show that to every e > 0 there is an no = no(e, k) so that if (34)

bl 2 k. We will only outline the fairly complicated proof. Assume that there is a sequence satisfying (34) for which g(m) < 2k for all m. We shall show that this assumption leads to a contradiction. We split the b's into five classes. In the first class are the b's which can be written in the form k+l

(35)

I-[e~,

e ~ > ( l o g n f ~,

l (log n) c~, 1 .~ i < k - 1 i=1

and where t can not be written as the product of two integers > (log n) ck (for otherwise our number would be of the first class). In the third class are the integers where all prime factors of t are less than (log n) ~' where r/1 = t/l(e) is sufficiently small. We can assume t < (log n)4Ckfor otherwise t would be the product of two integers > (log n) ~. Thus the number of integers of the third class is at most ~ ' 17k_ l((n/t)) where the dash indicates that t < (log n) 4c~and all prime factors of t are less than (log n) ~1. By a simple computation we have from (10) and t/1 = r/l(0

258 (37)

P. ERDOS ]~'Hk_l

[December

( t ) = (1 + o(1) n(l°gl°gn)k-2 ~' 1 < - -e n(loglogn) k-I (k - 2)!logn

10 (-k~

Thus by (37) the number of b's which belong to the first three classes is less than (e/2)(n(loglog n)k- l)/(k- 1)! log n) and hence by (34) there are at least n (log log n) k (1 + -~-) (-~-~-~.Vl~n

(38)

1

b's which do not belong to the first three classes. These b's can by (36) all be written in the form (Pk is the greatest prime factor of t) (39)

k t' 1-[ Pi,

t Pi > (l°gn) c~, 1 < i < k - 1, Pk > (logn) "', t' = - - .

Pk

i=t

In the fourth class are the b's for which t' < (log n) .2 where ~/2 =/72(?]1)

(40)

is sufficiently small. We shall now show that our assumption g(m)< 2 k for all m implies that the number N of integers of the fourth class is less than N'">2k,

2 ~ ' < p ~ < 2 l+a'

Denote by N(21,-.., 2k) the number of r's corresponding to the k-tuple (2t, ..., 2k). We shall show that (48)

N(21, "",2k) ~_ 2

xk=12i

('),1 [-I2~

~ 12--~k]2~+ 1Q

By the prime number theorem the number of primes p~ satisfying (47) is (1 + o(1))(2a'/2~1og2). Now we apply our Lemma with r = k and (49)

N, = (1 + o(1)) ~ 2 ~

> 2~k/2, 2~k > 21__(log n) "~.

We obtain by the Lemma by a simple argument that if (50) N(21 , "",2k) > (1 +

1

,

(1og2)-k 2-2k/2~>

)

2Zi=t2~ 1-I 2~ -1 2-ak/2~*' ',1=1

then primes pt(l~ p t2~ 1 < i < k exist so that the numbers (45) are all r,'s and we have assumed that such primes do not exist. Thus (50) is false or (48) is proved. (48) clearly implies (the dash indicates that 2 x~k--x2~ __6 n and 24~> ½(log n) ~t

260 (51)

[December

P. ERDOS

l= ~"N(2~"" • '2k)< ~'2Xk=12/( ~I=~2i ) -1

2--4k/2~'+1

i

By an elementary but somewhat lengthy argument (using elementary inequalities) which we supress we obtain that (51) implies (52)

1 < n/(log n) 1+ 3~2

if f]2 = ~2(/'/1) is sufficiently small. (52) contradicts (46) and this contradiction proves that (44) can not hold, which finally proves (41). The remaining b's are in the fifth class. By (41) these integers can be written in the form k (53) t' l-[ Pi, P / > (log n) ok, 1 _< i < k - 1, Pk> (log n) ~', (log n) ~ < t' < (log n) 4ck /=1 k 1 P/ can be (if t' > (log n)4~ then a simple argument would show that our t' I~/= written in the form (35) and hence belongs to the first class). By (38) and (41) there are at least

e n(loglogn)k-I (54)

-4- (k - 1)!log n

n > log---n

b's of the fifth class. To each such b we make correspond a (k + 1)-tuple (21, "",2k+1), 21 > "'" > 2k+1 satisfying 24` < p ~ < 2 1 + 4 ' ,

l ½(log n) %

1 < i < k , ½(log n) ~2
0 is a suitable positive constant. But at present I can not prove for l > 2 a result as sharp as (4) and (5).

REFERENCES 1. P. Erd6s, On sequences of integers no one of which divides the product of two others and on some relatedproblems, Izv. Inst. Math. and Mech. Univ. of Tomsk 2 (1938), 74-82. 2. P. Erd6s, On extremal problems of graphs and generalized graphs, Israel J. Math. 2. 3 (1964), 183--190. 3. P. Erd6s and O. R6nyi, Additive properties of random sequences of positive integers, Acta Arithmetica 6 (1960), 83-110. 4. E. Landau, Verteilung der Primzahlen, Vol. 1,203-213. 5. D. Raikov, On multiplicative bases for the natural series, Math. Sbornik, N. S., 3 (1938), 569-576. 6. A. St6hr, Gel6ste und ungel6ste Fragen iiber Basen der natiirlichen Zahlenreihe, I and ll, J. reine u. angew. Math., 194 (1955), 40-65 and 111-140; see p. 133. TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY, HAIFA, ISRAEL

INDEPENDENT SETS IN REGULAR GRAPHS BY

M. ROSENFELD* ABSTRACT

Lower and upper bounds for the maximal number of independent vertices in a regular graph are obtained, it is shown that the bounds are best possible. Some properties of regular graphs concerning the property .~ defined below are investigated. Introduction. In this paper we are interested in independent sets in regular graphs. In §2 we give bounds for the maximal number of independent vertices in a regular graph G. (G will always denote a graph without loops and multiple edges). It is shown that these bounds are best possible. It seems true that each value between the bounds is obtainable. In §3 we define the property ~ for graphs: we say that a graph G e . ~ if every vertex of G belongs to a maximal independent set of vertices in G. In some cases conditions are given under which G e . ~ . In §3 we define a class of graphs called homogeneous for which it seems to be interesting to investigate their properties and structure. 1. Definitions and Notations A graph G will be called regular of degree m if every vertex is incident with exactly m edges. We shall denote such a graph by G(n, m) where n is the number of vertices in G. It is evident that such a graph exists iff n > m and n ' m - 0 (mod 2). a(M) will denote the number of elements of the finite set M. 6 will denote the complementary graph of G. The components of G are its maximal connected subgraphs. A set R = { a l " " ak} ~ S (S the set of vertices of G) will be called a representing system of the edges of G if every edge of G is incident with at least one vertex from R. #(G)will denote the minimal number of vertices representing the edges of G. /7(G) will denote the maximal number of independent vertices in G. v(G) denotes the number of edges in G. M ~ S. [M] will denote the subgraph spanned by M. M, N ~ S, a MN-edge is an edge whose one endpoint is in M and the other in N. [M, N] will denote the subgraph whose vertices are M u N and the MN-edges contained in G. a, b e S, (a, b) e G denotes that a and b are incident in G. Cn will denote the complete graph with n vertices. Received January 20, 1965. * This paper is to be a part of the author's Ph.D. thesis written under the supervision of Prof. B. Griinbaum at the Hebrew University of Jerusalem.

262

1964]

INDEPENDENT SETS IN REGULAR GRAPHS

263

[fl]* will denote the smallest integer not less than ft. 2. THEOREM I. Let (3 = (3(n, m) be a regular graph of degree m with n vertices. Denote by Ft(G) the maximal number of independent vertices in G then:

k+2 b) fi(G) >_-

3

n=k(m+l)+m;

m>2

-~

(3 = G(2n + 1,2n - m); m < n;

> 2n + 1

, n a . ot er ase These bounds are best possible in the sense that for each pair of integers n, m such that m < n and n • m -= 0(rood2) they are obtainable. Proof. Let S be the set of vertices of G. Let A c S be an independent set of vertices in (3 and ~(A) = fi(G). Since A is a maximal independent set of vertices, each vertex of S - A is neighboring with at least one vertex of A. The number of different edges having one endpoint in A is/~((3) • m hence: 1) ~ ( G ) ' m >=n - ~ ( G ) ~ ( G ) >

= [ ~ ] *

(/7(G) is an integer !)

2) v((3) = n . m =~ Tn ' m > ~ ( G ) " m ~ [ 2 ] 2

>/~(G).

Let a ~ A, The m different edges whose one endpoint is the vertex a have their second endpoint in S - A. Therefore 3) ~(S - A) = n - ~(G) > m ~/7(G) < n - m. To show b) we need two Lemmas:

L~CIMA I. Let G = G ( K ( m + l ) + m ; m ) , (i.e.~(G)_>_

~

m>2~/~(G)>K+I +1).

Proof. (Observe that m must be even otherwise the graph does not exist). To prove the lemma we use induction on k. For k = 1: Since G = G(2m + 1, m) ~ G = G(2n + 1, m) and since an independent set of k vertices in G form a complete k-subgraph in G it will suffice to show that (~ must contain a C 3 . Let a ~ S be any vertex in G. a ~ {a~ ... am} = A. If a is not contained in a C3 A must be independent. S = {a} ~ A@B. e(B)= m. This implies that each at E A is an endpoint of m - 1 edges whose second endpoint is in B.

264

M. ROSENFELD

v(G) = m 2 +

[December

_• ~v([B]) = m 2 + -~- - (m + (m m

m

1)m) = -~-.

Since by our assumption m > 2 and m is even v ( [ B ] ) > 2. Let bi--', bk, bi'-"} b[, (it is possible that bi = b~ but bk ~ bk) bi is neighboring with some ai ~ A otherwise bi @ A would be an independent set with m + 1 vertices which is impossible. If a,-~bk a i - - { B - b k ) ai~b'~ and ai~b'k and [a,b~b'k]=C3. If a , ~ b k then [aibibk] = C3. Suppose ko > 1 is the smallest positive integer for which the lemma does not hold. Let us denote by Go the graph satisfying

Go = G(ko(m + l) + mlm) /7(Go) = ko + l.

(1)

G o cannot be connected: if G o is connected by a theorem due to Brooks [1] G o is m-chromatic. Hence Go = 25 (b At i=1

where Ai is the set of vertices colored with the i-th color. max~(Ai) >

ko(m + 1)+ m m

> ko + 1

Since A i is an independent set of vertices in G o this is a contradiction to our assumption (1). Hence Go must have at least two components: (2)

G O ---- G 1 (~) G 2 .

Let us consider the two possible cases: i) G 1 = G(kl(m + 1) + r I , m) G2 = G(k2(m -I- I) + r2, m) kl + k 2 = r 1 q-r 2-m

ii) G l = G ( K l ( m + l ) + m , m )

ko,

r2, r 1 ~0

G2 - - G ( K 2 ( m + l ) , m ) .

CASE (i): /7(Go) =/7(G1) +/7(G2) > K1 + 1 + k 2 -1- 1 = Ko + 2 which contradicts the assumption (1) hence this decomposition is impossible. CASE (ii): By the induction hypothesis (K 1 < Ko)~(G1) >K1 + 2 /7(Go) =/7(G1) +/7(G2) > K1 + 2 + K2 = Ko + 2. This shows that /7(Go) = K o + 1 is impossible and the proof of the lemma is completed.

5m

LEMMA II. Let G = G(2n + 1 ; m) and suppose that --~ > 2n + 1 then G

contains a Ca.

,g.

1964]

INDEPENDENT SETS IN REGULAR GRAPHS

265

Proof. Suppose that G does not contain a C 3. Let a s ~ (ba, b 2... bin} = B ba -~ ( a l ' " am} = A. I f G does not contain a Ca, A and B are independent sets of vertices and A n B = 0. Therefore G = A ® B • C where C = (ca ' " c,} and m by our assumption r in an odd integer and r < -~. Let c e C and suppose that c has r a < r - 1 neighboring vertices in C. Suppose furthermore that c

"-~ (aft...

a~

bkl"'" bk)

j + k = m - ra.

Without loss of generality we may suppose that j > k. Since B is independent bg, has m neighboring vertices in A @ C. Now c has r a neighboring vertices in C; then if G does not contain a C a bk~ has at most r - r~ neighboring vertices in C. Therefore bk~ has at least m - r = r~ neighboring vertices in A. A contains m - j vertices that are not incident with c. j+k=m-r

a

j>k

rn + r a 2

m-j r, m - r + r 1 > -~ + r 1 > rx > m - ] ; this means that bk must have at least one neighboring vertex from a h ... aij and G o would contain a C3, a contradiction to our assumption. Therefore we conclude:

i) c~ai~c~{B}. ii) c , c ' e C , c ~ c ' ,

c-~{A}~c'-~{B}. t

t

m

For if we had c -~ {a,l ... aij } = A c ~_ A, and c' -~ (a,, ... a,k } = Ac,, k = m - rl > - ~ , m j = m - r a > ~- A c N A t , ~ 0 and Go would contain a C 3 (N l,r~ have the same meaning as in the preceding paragraph). Denote by Ca = {c a ... c~}_ C and C B = (c] ..-c~,}_ C those vertices of C having neighboring vertices in A or B respectively. Because of i) and ii) C = CA • C~ and CA and Cs are independent sets in Go. I f Go does not contain a Ca we conclude from the above discussion that Go contains only four types of edges: (1) CaCB-edges (2) ACa-edges (3) BCB-edges (4) AB-edges. Henceforth to complete our proof it will suffice to show that r i B , CB] # v i A , Ca]. (This means that if G Odoes not contain a C3 it cannot be regular). Since r is odd and j + k = r we may suppose tbat k > j . L e t r~ = drc~(Ci) we have rt CaCB-edges in Go, j . m - Z~=a r, ACA-edges and k. m - ~ = x r, BCs-edges. Since k > j k " m - ~,r~ = r i B . CB] > j " m -- ~,~=~ r, = v [ A . C4] Q.E.D.

~=1

Lemmas I and I I give the justification of the modification of the lower bound for/7 (G) given in (b).To complete the proof we will construct for each given admissible m, n a regular graph in which we obtain the bounds.

M. ROSENFELD

266 (a) G(2n, m)

m -~- take in the first case G = Cm+l @ ""Cm+l@Ga and in the second case:

G= Cm+l@"" @Ge Where G a - - G ( m + r + 1, m) is the graph constructed in (a) (/7(Ga)= 2 ) a n d

Ge = G(m + r + 1, m) is the graph constructed in (e). One easily verifies the equality ~ ( G ) = [ - ~ + 1 ]* . If m + r + l is odd but m + r + l < - ~ - ,5r mmust be even. Let 2m + r + 2 = 3 k + J o 0 < J o < 2 ( 2 m + r + 2 is even!). I f j o = 0 k is even. Let Ci . . .{c~ i 1 r, G(n, m) must contain a C,+ x. (This result can be obtained from Tur~m's theorem). A slight modification of this result can be obtained from lemma II: If

G = G(u(m + 1) + m,(u - 1)(m + 1) + m)

then G contains a C,,+2. 3. DEFINITION A graph G will have the property ~¢'(G e oW) if every vertex in G is contained in a maximal independent set of vertices with p(G) vertices. In this section we shall investigate the property ~ in some extremal cases of regular graphs. For this we need few more definitions: 1) With each vertex of G(n, m) we associate a (m/2)-tuple o f integers ordered by increasing magnitude and defined as follows: with each two edges incident with the vertex in consideration associate the length of the shortest circuit containing them. If such a circuit does not exist the number associated will be + oo. We denote by z(a) the (m/2)-tuple associated with the vertex a, and call it the type of a. It is obvious that a necessary condition that there exists an automorphism of the graph that carries a to b is z(a) = z(b). 2) A regular graph will be called homogeneous if all the vertices in each component have the same type. Examples of homogenous graphs are circuits, complete graphs and point symmetric regular graphs. A homogenous graph need not be point symmetric, see for example the graph constructed by B. Griinbaum in [3] L E n A 3.1. G i s a g r a p h . d ( a ) < m

VaeG~Ft(G)>

Proof. We use induction on n. (~(G) = n). For " s m a l l " n' s the lemma is obvious. Let a e G. a ~ {bl'." bm, }m' < m. e(S - {a b l ' " bin,}) = n - (m' + 1). The graph G' = I S - { a b l ." bm,}] satisfies the conditions of the lemma hence m+--1

->-

-1

therefore a maximal independent

set from G' together with a is an independent set A with e(A) > THEOREM 3.1. G = G(n • m).

=[

Ft(G) [ m +

1J

,Ge

~,'g'.

270

M. ROSENFELD

[December

Proof. The proof is a direct consequence of lemma 3.1. since we have shown

that any arbitrary chosen vertex belongs to a maximal independent set. n

THEOREM 3.2. G = G(n,m) ~ ( G ) = ~ = ~ G e ; , ~ mined, m+ 1

and G is uniquely deter-

Proof. P. Tur{m in [4] proved that in this case G is the direct sum of complete m + 1-graphs, hence the theorem follows.We give here another proof of the theorem. n We use induction on k = For k = 1 the theorem is obvious since in this m+l" n c a s e G = G m + 1. Let G = G ( n , m ) and ~(G)= m + 1 = k > 1 be given. If G is connected by a theorem of Brooks ['1], G would be m-chromatic, G = ~ = t (3 At, At is the set of vertices colored " i " . n

n

max ct(ai) ~_ - - > m m+l

n But Al is independent, in contradiction to the assumption ~(G) = m +-------T'" Hence G = G t 0) G2 where Gt = G(nlm), G2 = G(n2, m). n t + n2 = n Suppose

na m+l

and

is not an integer =~ =

Hence: m n~ + 1 - k~ < k

---:--:,.

m n2 +

/~(G) = g(Gt) + g(G2). n2

m+l

+

i =

is not an integer by Theorem I: >

=

m+l

+

1

>

~

re+l"

k2 < k and by the induction Hypothesis

they are the direct sum of complete m + 1-graphs. This means that G is point symmetric :~ G e ~¢t~. THEOREM 3.3. Let G = G(n,m); m > ½n; p(G) = n - m then: 1) G e ~ if and only if n - m/n and G is homogeneous. In this case G is uniquely determined and point symmetric. 2) G ¢ ~ ' if b does not belong to a maximal independent set of vertices while a does then z(b) < z(a). (The types are ordered lexiocographically). Proof. 1) Suppose G e ~ .

Let A 1 = (as,..., an-m} be a maximal independent Therefore if A2 = {a~.-. a,'-m} is a maximal independent set different from AI we must have As N A2 = ¢. Now if G e ~ each g e G belongs to a uniquely determined maximal independent set of vertices, hence G is the direct sum of independent sets of vertices and each vertex is connected by an edge to all the vertices not belonging to the independent set including it. This means that n - m/n. Since the complementary graph of A is set. H e n c e

a, ~ { S - & } .

1 _< i -< n - m.

1964]

INDEPENDENT SETS IN REGULAR GRAPHS

271

easily seen to be the direct sum of complete n - m graphs, G is point symmetric and therefore homogeneous. It is then obvious that if n - mXn :~ G 6 ~ . 2) Let a e A , A a maximal independent set =>a ~ {S - A). Let B be the set of all vertices that are not joined by edge to b (including b) :~ct(B) = n - m. Since each a e A is joined by an edge to S - A and b e S - A. B ~ A = O. Hence: G = C (3 A ~ B. Denote by l(xay) the length of the shortest circuit containg the edges (ax) and (ay) (in the sequel it will be shown that l(xay) is finite). Let us calculate z(a) and

,(b). Put {cl ... cs} = C

cicj e C and

s = 2m - n.

(cicj) e G

(c~cj) ~ G

l(ciacj) = l(cibcj) = 3

l(ciacj) = l(cibcj) = 4.

l(a~bc~) = 3 this contributes ( n - m)(2m - n) times " 3 " to z(b). l(a~baj) = 4 this contributes ½(n - m)(n - m - 1) times " 4 " to z(b). Sin ce b does not belong to a maximal independent set:

v[B] = r >=1 Suppose therefore (b~bj) e G :~ 3 c', c" e C ^ (bid), (bjc") ¢ G. :~ l(biabj) = 3,

l(b~ac') = 4, l(bjac") = 4. Since riB] = r it is easily seen that we have 2r triangles of type l(abc) more than of type l(cab)while only r triangles of type l(bab)more than of type l(aba); this shows that in z(b) we have r " 3 " more than in z(a) ~ z ( b ) < ~(a). This proves also that if G is homogenous we must have n - m/n and G e ~ . This completes the p r o o f of the theorem. THEOREM 3.4. G = G(n, m) /7(G) = 3. I f a does not belong to a maximal independent set of vertices in G while b does then z(a) < z(b). Proof. Observe first that if m < 2 ' G e . ~ , hence we will suppose that m > n =

2"

Let a ~ { x l " " X m } =X,,, a-~,{yl...y,_m_l}= Y,,. 1) G = { a } ~ X . @ Y. 2) a does not belong to a maximal independent set implies Y. = C._ m_ i. 3) The number of different triangles containing a is v[X.].

v[X']=n'm2

{m+

(n-m-1)(n-m-2)2

t-(n-m-1)(2m-n+l)}"

Let b ~ { r l . . . r m } = R b b + ~ { p t . . . p . _ m _ l } = Pb. G = {b} ~ R b G P b. Since b belongs to a maximal independent set Pb # Cn-m-t

the number of

272

M. R O S E N F E L D

different triangles containing b is v[Rb]. It will therefore suffice to show that V[Rb] < v[Xa]. Suppose that in [Pb] r edges are needed to complete the graph [Pb], the two endpoints of such an edge are connected by an edge to vertices in Rb. Hence for each "missing" edge in Pb we have two "additional" PR-edges:

V[Rb] _ n -- m 2

{

mq-

(n -- m - - 1 ) ( n - - m - - 2) 2

r+(n--m--1)(2m--n+l)+2r

}

Since r > 1 =~v[Rb] < v[Xo]. This completes the proof. REMARKS. 1) G = G(n, m) f~(G) = 3 G is homogeneous =~G ~ • . 2) In the general case we do not know when G E ~¢~; it is obvious that: G e J f ¢> c3 Ro = ~ (R~ runs over all the minimal representing systems in G) but this is not a useful criterion.

REFERENCES 1. R. L. Brooks, On coloring the nodes of a network. Proc. Cambridge Philos. Soc., 37 (1941), 194-197. 2. P. Erd6s and T. Gallai, On the minimal number of vertices representing the edges of a graph Magyar Tud. Akad. Mat. Kut at6 Int. K6zl., 6 (1964), 181-202. 3. B. Grtinbaum, A problem in graph coloring. (Unpublished). 4. P. Turfin, On the theory of graphs, Coll. Mathematicum, 3 (1954), 10-30. THE HEBREW UNIVERSITY OF JERUSALEM