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

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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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. ;I Fig. 4. d, E D - D,, and f,E A nB. The cycle in the thick line is longer than L 46 B.

In [7] we conjectured that K,” can be decomposed into hamiltonian cycles if and only if (9/n is an integer and proved this conjecture for n a prime. 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.

This was to be proved. Part I1 (First version). (A) We start with a graph G" for which h ( G " )3 cn2, fix a c"< c and then a C ' E (c", c) and a p > 0, which is much smaller than c". Using the first part we may assume that the edges are regularly distributed. We may repeat part (A) of the proof of Theorem 7 replacing e( ) by h( ) and c by c'. Then we may assume that G' = G", but have to decrease c': replace the original condition by condition h ( G " ) s c"n2. How we define the graph R k as in the beginning of (B) of the proof of Theorem 7.

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

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