By B. Bollobás (Eds.)
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Extra info for Advances in Graph Theory
Hence the second assertion of Theorem 1 is trivial. To prove the first assertion assume indirectly that G" contains no L +E'""] and e ( G " ) > q(n, L ) . We shall show that this is impossible if c > 0 is sufficiently small. By Lemma 3 and Lemma 4 we may and will assume that 6 ( G n )2 (;-&r-')n , R = gr, and G " S K3(9r,9r, 9r). Applying Lemma 6 with d = 2, we obtain a partition ( A l ,A J , satisfying (i)-(iv) of Lemma 6. For the sake of convenience in the sequel a subset H of the vertices of G" and the corresponding spanned subgraph may be denoted by the same letter.
C,@S, can be decomposed into n hamiltonian cycles. 13. Remark. 3 since Hamiltonian decompositions of graphs, directed graphs and hypergraphs 23 K,,, = K,@S,. If r is odd. 12. 2. Finally KrxZp = ( K r x 2 ) @ S p U,C$:@Sp= = UI,lC$$). And thus if n ( r - 1 ) is even K,,, can be decomposed into hamiltonian cycles. 14. Theorem (Laskar ). Cr€3C, can be decomposed into n + 1 hamiltonian cycles if n is odd or r is even. 15. Conjecture. C, €3 C, can always be decomposed into n + 1 hamiltonian cycles.
An interesting feature of the result is that the value does not really depend on j and t. Theorem 7 . Let j , r, t be natural numbers, let k = 2j+ 1 and let c > 0 . If e ( G " )2 cn2 and G" does not contain a C k ( t ) ,then G" contains a K2(r,m ) , where = 2 2 r - l c r n+o(n). Proof. We shall show first that if instead of a C " ( t ) (and so a fortiori a C k )we prohibit all odd cycles, then G" contains a K2(r,m ) with = 22'-L c 1n + o ( n ) , but if E > O then G" need not contain a K2(r,m ') with m ' = ( 2 2 r p 1 c+r~ ) n + o ( n ) .
Advances in Graph Theory by B. Bollobás (Eds.)