B. Bollobás (Eds.)'s Advances in Graph Theory PDF

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Hence k e(G")s i=s+2 (d,n)m+(d,n)mc(dn)(n-u). 38 B. Bollobas, P. Erdos, M. Simonouits, E. Szemerkdi where a = /1 = rIJs s + , V , j = ( s + l ) m . To obtain an upper bound of d in terms of a, we apply Lemma 5 to the bipartite graph determined by the classes Uzs,ss+lV , (=first class) and V, (=second class). We find that G" I K z ( r , t ) with t = ( 1-o(l))drnra-(r-l1. (7) By the assumption G " 3Kz(r, 22'C'c;n) and by (7) (8) drnrC1a-(rCIJ< ( 1+0(1))2~'-'c;. Let us assume that d > 2c, (this will be shown later).

8. Conjecture (Kotzig [12]). C, x C, x C,,, can be decomposed into hamiltonian cycles. 9. 8. Koester (personal communication, 1977) has proved that C, x C, x C, can be decomposed into hamiltonian cycles. In fact he informed me that the problem of the existence of a decomposition of C, X C, X . . x C, ( n times) was posed by Ringel [16, Problem 21 as the existence of a decomposition of the 2n-cube (2n-dimensional Wiirfel) into hamiltonian cycles. 6. Very recently I learned that the existence of a decomposition of C, X C, X C, into hamiltonian cycles was proved by M.

For each vertex b , E B - A there is an interval on L consisting of vertices bl, cl, f , , c2, f 2 , . . ,fi ,, c,, a,, where c, E C, f, E A n B and Q, E A -B. There are s such intervals and so the vertices of D also form S intervals, some of which may be empty. Let I, be the length of the ith interval. Then CS=,,lz= ID\ = 2 k - r - s. The set D - D,, does not contain adjacent vertices since otherwise G contains a longer cycle than L, as shown in Fig. 4. Consequently s ID,,[2 1[ + I z ] 51 ' 3 1=l (I, - 1)= k - s - ir.

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

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