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

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Keywords » program of Graph thought for effective research - Finite aspect research - Meta-heuristic Algorithms

Related matters » Computational Intelligence and Complexity - Computational technological know-how & Engineering

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Given c, 0 < c < 1, and natural numbers d and R, there exist M = M ( c , d . R ) , 6 = 6 ( c , d , R)>O, and no= n(c, d , R ) such that if n > n o , e ( G " ) > ( I - l / d - S ) $ n 2 and K,+,(R,, . , R)@ G", then the vertices of G" can be divided into d classes, say A,, A,, . . ,A,, such that the following conditions are satisfied. (i) ( n , - n / d l < cn, where n, =lA,l. (ii) The subgraph G, = G"[A,] of G", spanned by A,, satisfies e ( G , )< cn? (iii) Call a pair {x, y } of vertices a missing edge if x and y do not belong to the sume class A, and xy is not an edge of G".

Remark. Theorem 9 is sharp, apart from the value of I(c) which is probably O(c I / * ) . This l ( c )= O(c- ' I 2 ) would be sharp if true. To see this put n = (21+ 3 ) , and G" = C" " ( m ) . If c = (21+4)C2,then more than cn2 edges must be omitted to turn G" into a bipartite graph and C" does not contain Ck if k is odd and smaller than 21+3. Proof. Our proof consists of two parts. We shall give two versions of the second part. Extrernai graphs without large forbidden subgraphs 39 Part I. Let c' < c be fixed.

Exactly as in the proof of Theorem 7, we can prove that G " contains at least c , n J cycles CJ,where c , > O is a constant. Applying the theorem of Erdos on hypergraphs [4], we obtain j sets X I , .. , X I with IXll= T+m, such that if x , E X , , . . , . ,x ~ , )is a cycle of G" (we consider here the hypergraph whose hyperedges are the j-sets of vertices of j-cycles in G"). Unfortunately the cycles will not determine a CJ(T), since the permutation i,. . , i, may differ from j-tuple to j-tuple.

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


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