A c t a M a t h . A c a d . Sei. H u n g a r .
40 O---4), (1982), 201--208.
3K~-DECOMPOSITION OF A GRAPH A. BIALOSTOCK...
160 downloads
516 Views
338KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
A c t a M a t h . A c a d . Sei. H u n g a r .
40 O---4), (1982), 201--208.
3K~-DECOMPOSITION OF A GRAPH A. BIALOSTOCKI and Y. RODITTY (Tel-Aviv)
1. Introduction
Graphs in this paper are finite, have no multiple edges and loops. DEFINITION 1. A graph G=G(V,E) is said to have an H-decomposition if it is the union of edge-disjoint isomorphic copies of the graph H. Necessary and sufficient conditions for H-decomposition have been determined mostly for the complete graph Kn, see [1, 2], but also for complete bipartite [2] and complete multipartite graphs [2, 5]. However only for particular graphs H. Recently Y. Caro and J. SchSnheim considered H-decomposition of a general graph where H is 2K~ or Pe (two-bars or a path of length 2). This problem was completely solved [3, 4]. This paper determines the graph G which have 3K~-decomposition. It is proved that the necessary conditions are also sufficient excluding a list of 26 graphs. 2. Preliminary results
The following two conditions for are obviously necessary: (1)
G=G(V, E) to have a 3K2-decomposition
E(G)=3k,
(2) deg
(v)--6. In the course of this paper we shall deal with the sufficiency problem. DEFINITION 2. If U is a subset of vertices of the graph will denote the degree of v in the graph induced by U U {v}.
G(V, E) then degv(v)
DEFINITION 3. Let G=G(V,E) satisfy (1) and (2) for a certain k. Denote VI={v[vEV(G), deg(v)=k}, and its cardinality by ~. DEFINITION 4. Let G=G(V, E) be a graph and H a subgraph of G. Denote by G',,,H the graph whose set of vertices is the same as that of G and its set of edges is the set E(G)\E(H). DEFINITION 5. Let G=G(V, E) satisfy (1) and (2) for a certain k. Define X = ~/~ degv~ (v), Y= ~, d e g v \ v I (v), where V1 is as in Definition 3. vEV a
vEV a A c t a M a t h e m a t i c a A c a c l e m i a e S e i e n ~ i a r u m H u n g a r i c a e 40, 1982
202
A. BIALSTOCKI A N D Y. RODITTY
THEOREM l. Let G=G(V, E) satisfy (l) and (2)for k. Then, (a)
X ~ 2(~-3)k
(b)
a=7.
~3,
PROOF. Summing degrees in G implies
(3)
X + Y = ~k.
Counting edges implies X ~ - + Y N 3k,
(4) or
(4')
X + 2Y ~ 6k.
Subtracting (4) from (3) implies (a). Subtracting (3) from
(5)
(4') implies
Y 3 then a, bEV(3K2). PROOF. The only graph that does not contain 2//2 are K3 and a star. But K3 and a star do not satisfy- (1) and (2) for any k. Hence our graph contains 2K2. If there are a, bEV(G) such that d e g ( a ) ~ 3 or deg(b)=>3 then there exist vertices x,yr b such that (x, a), (y, b)EE(G). In any case we shall take 2K2 as (x, a) and (y, b). Let zl, ..., z, (n~2) be the vertices left in G. If there is any edge (z~, zj), we are through. Otherwise all the vertices zl, i r . . . . , n are adjacent only to {a,b,x,y}. Let G2=G2(Vz,E~,) be a graph such that V2= {a, b, x, y} and E2={(s, t)EG]s, tEV2, (s, t);~(x, a), (y, b)}. Denote fl=]E(G2)]. Then obviously 0-