2-Matchings and 2-covers of hypergraphs

A c t a Mathematieae Academia Scientiarum Hungaricae Tomus 26 ( 3 - - 4 ) , (1975), 433---444.

2-MATCHINGS AND 2-COVERS OF HYPERGRAPHS By L. LOVA_SZ (Budapest)

w O. Hypergraphs By a hypergraph ~ we mean a finite collection of non-empty finite sets; "collection" means that the same finite set may occur more than once. The elements of a hypergraph are called edges; the elements of edges are called vertices (this way we exclude "isolated vertices", i.e. vertices not occurring in edges; these vertices would be insignificant in our discussion). The set of vertices of the hypergraph J4v will be denoted by V(Jd). We need some operations defined on hypergraphs. Removing an edge means removing this edge (and, of course, all points which become isolated by this). Removing a vertex x E V(Ne) means the removal of all edges containing this vertex. Doubling a vertex x means the addition of a new vertex x' and of a new edge E ' = E -{u}U{x'} for every EEJF, such that xEE. Finally, cutting off a vertex x can be carried out if {x}E 24~ then it results in the hypergraph. {E-{x}'. EEJ~ ~ }. A hypergraph obtained from A" by removing edges (points) is called a partial (induced partial) hypergraph. The hypergraph obtained by removing the point x is denoted by AV-x.

w 1. Ma*chings and covers of hypergraphs; results Let all' be a hypergraph. By a k-matching o f ~ we mean a collection A/" of edges of ~ (the same edge of ~ may occur in JV" more than once) such that each point of ~ occurs in at most k members of J~. The size • of the k-matching dr" is the number of its edges. A k-cover 3- is a collection of points of x4~ such that each edge contains at least k of them. The number ~ of elements of Y is called the size of ~7. By a fractional matching we mean a system of weights hE_> 0 associated with the edges E of A" such that z~ ne 0 associated with the points x of o~ in such a way that ~ ' tx--> 1 for each edge E. The size of a fractional matching (cover) is ~ n e ( ~ t x ) . xEE

E

x

If d is a k-matching then denoting by N~ the number of times E occurs in d , 1 nE=~-NE will be a fractional matching. We will denote this fractional matching X by -~-. If ~ is a ki-matching (i= 1, 2) then the collection ~/~+dg~2, in which an edge E occurs the number of times it occurs in ~ plus the number of times it occurs in ~/~, is a (kl+k~)-matching. Similar definitions and observations hold for covers. 14"

Acta Mathematica Academiae Sc~entiarum Hungaricae 26, 1975

434

L, LOVASZ

We denote by vk(g/g') the maximum size of k-matching and by v*(Jg) the maximum size of a fractional matching. The minimum size of/c-covers and fractional covers is denoted by %(Jr and -c*(o~f), respectively. We remove the argument 24~ if there is only a single hypergraph in consideration. Also we set r~=v and v~= v. It is clear from the definitions that (1)

V ~ __V~-< V*

and

(2)

qT* ~ S~k-2). Consequently, %(G)=Zz*(G). PROOF. Let f l be a k-cover of G. Set X~= {rE V(G): v occurs in J i times}. Clearly we may suppose X~= | for i>k. Define J~ to consists of 2 copies of v for vEXk 1 copy of v for

vEX1U...UXk_I.

Also let ~r consist of k - 2 copies of v for vEXk i - 1 copies of v for vEX~, 1 < : i < : k - 1 . Then clearly f l = J l + J ~ . We show J l is a 2-cover while J~ is (k-2)-cover. For let {x, y}EE(G). If none of x,y belongs to X0, then clearly {x, y} is covered twice by J l . If, say, xEXo then since J covers {x, y} k-fold, we must have yEXk and again, J~ contains y twice. On the other hand, we show f12 covers {x, y} (k-2)-fold. This is clear if one of x, y belongs to Xk. If they do not belong to Xk then J l covers {x, y} exactly twice and since J covers {x, y} at least k-times, J ~ = J - g r must still cover it at least ( k - 2)-times. Let f l be a minimum k-cover. Then J = J l + J 2 where j~ is a 2-cover and j~ is a (k-2)-cover. Hence

As trivially "%+%-z=>zk we have %=%+zk-2 and consequently %a=k.z2. Now by the remark after the definition of v* and ~* we have a k o such that z2~o=2ko-z*, implying %=2z*. This completes the proof of the lemma. LEMMA2. Let ~ be a hypergraph with maximum degree 2. Then each k-matching of ~ is the sum of a 2-matching and of a (k-2)-matching. Consequently, v2k=k, v2 and v~=2v*, The proof is analogous to that of Lemma 1 and is omitted. Now we turn to the proof of Theorems 1 and 2. PROOF OF THEOREM1. Let ~z~ be a counterexample to the theorem with the least possible number of vertices. Let x 6 V(~). Since J(~-x also satisfies the assumption of the theorem and has less points than Yf, it will have (4) Since (5)

v2(Jr~- x) = % ( J g - x). v2(

)
~2(~)-2, we have by (4) and (5), v~ ( a V - x) = ~'~( J ~ - x) => ,~ ( ~ ) - 2 > v~ (J~) - 2 or (6) v2(Yt~ x) _-> v~(Y~) - 1 for any x. Let us double now as many vertices of ;/f as possible to get a hypergraph J/f' with v~(o"/g')= v2(Jf). Let T denote the set of those vertices of W which have not been doubled. We claim T meets every edge of W ' in at least two points. Suppose first there were an edge E6E(,Cf') with EO Tc= {x}. Let us remove the points of E and one copy of each other doubled joint. The remaining hypergraph is isomorphic to ~"/g or W - x and contains a 2-matching ~ of size v~,(Yg)- 1. Now d4+{E}+{E} is a 2-matching of ~/g' of size v2(3r 1 which is a contradiction. Thus we have shown that T is a 2-cover and, consequently, z~(Yt~ ]Tt. Since ~/g is a partial hypergraph of ~'g, we also have (7)

z~(Y~) ITl(v=(~)+ 1) xET

and thus by (8), ITI(v~(W)+I) 1+[s odd components, a contradiction with ( . ) . Let R be a type II set. Then no maximal set T c R , TC~g can be of type I or II since ~ is connected. Thus they all have to be one-element sets occuring with multiplicity 1. Suppose there are a type I][ sets in Jg. By the above, they are disjoint and their elements form one-element members of JCL Let b, e, d denote the number of points outside R~U...UR, occuring with multiplicity 2, 1 and 0 in ~ . Removing the d points not occuring in d{ the remaining graph will have a+b odd components. Hence a+b