By B. Bollobás (Eds.)
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Additional info for Advances in Graph Theory
It is easy to see that if G, H are graphs containing no L and no K2(1, cm/2), then G + H contains no L + E'""'. 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.
1. The complete graph K, has a l-factorization for each positive even integer n. Using the property that K , is an induced subgraph of K,+,, one quickly obtains the following. 2. For each positive odd integer n, the edges of the complete graph K,, can be partitioned into n sets each being a matching with ( n - 1)/2 edges. Let H be a subgroup of the symmetric group S, with index t. Let H , = H , H 2 , . . , H , be an enumeration of the right cosets of S,, with respect to H. Let 51 Chromatic index of the graph of the assignment polytope G , ( H i ) denote the subgraph of G, induced by the vertices in Hi(1s i c t).
Here I want to prove two theorems concerning the case h = 3. 2. Theorem. If K: can be decomposed into hamiltonian cycles then K:, also can be decomposed into hamiltonian cycles. (This proof was obtained with D. ) In order to shorten the writing, I will write a hamiltonian cycle as E,,. . E,, . . ,En where EL= (xE,y,, IC,+,). Let the vertex set of K:, be X U X’ with = [X’l=n. With each hamiltonian cycle of the decomposition of K: we associate 4 hamiltonian cycles of K : , in the following manner: 1x1 (i) if n is even, we associate with (x,y,x,)(x,y,x,), .
Advances in Graph Theory by B. Bollobás (Eds.)