By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Proof. By Lemma 2 every vertex of R has degree at least m - k - 1 in W. Furthermore no vertex of (B - A ) U Do is joined to any vertex of A f l B. Hence each vertex of ( B - A ) U D , has degree at least m - k - s 3 m - k - ( 2 k - r ) ~ m-3k+r in W. Thus R U ( B - A ) U D , c M and, by Lemma 5, ( RU (B - A ) U D o l s k. 17 Let now p=max{t: deg, ( w k + , ) a m - 3 k and G[wl, w2,.. , wk+t] contains 2t independent edges}. Lemma 6 implies that p 3 0 . Since 2p independent edges have 4p vertices, we have 4p s k + p so 0 s p s i k .

Lemma 2. q ( n + 1, L ) - q ( n , L ) a n / 2 . Extrernal graphs without large forbidden subgraphs 31 Proof. Let q(n,L)=n,n,+ex(n,,L)+ex(n,,L), where n,Sn,. Then q ( n + 1, L ) z ( n ,+ l)n,+ex ( n ,+ 1, L ) + e x ( n z ,L ) a 4(n, L ) + 4 2 . 2 of [ I ] . Lemma 3. Given c,>O there exists c,>O such that if e ( G " ) > q ( n ,L ) then G" contains a subgraph GP satisfying p 3 c2n, e ( G P > ) q ( p , L ) and S ( G P > ) (4- c,)p. Lemma 4. There exists a constant c,>O such that i f 6 ( G n ) 2 ( $ - & ) n and K = K3(9r,9r, 9r) c G", where r = IL(, then G" contains an L + E' with t 3 c,n.

6 cliques (maximal complete subgraphs) each having six vertices. For more information on these and other theorems concerning G, one can consult [31. 2. The chromatic index of G,. Let G be a graph and let A be the largest degree of a vertex of G. A matching of G is a subset F of the edges of G such that no two edges of G have a common vertex. If the matching F has the property that each vertex of G meets an edge of F, then F is called a perfect matching or l-factor of G. The chromatic index q ( G ) of G is the smallest integer t such that the edges of G can be partitioned into t matchings.

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Advances in Graph Theory by B. Bollobás (Eds.)


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