By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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**Additional resources for Advances in Graph Theory**

**Sample text**

Join x' to y t and y2. Denote by G ' the graph constructed in this way. By construction G' is a bipartite graph with vertex classes A and 2 = ( W- V ( P ) )U {x'}. We shall show that G' contains a Hamiltonian cycle. Since x 1 is adjacent to y, and x2 to y 2 , a Hamiltonian cycle of G' can be pulled back to a Hamiitonian cycle of G: all we have to do is replace the path x l x ' x z by x , Px2. 71). Somewhat unnaturally we state this result as an assertion about the graph G' at hand. Let d , S d, s .

3. Theorem. Zf n = 2 mod (3) K: can be decomposed into hamiltonian cycles. E. Brouwer (personal communication, 1976) for the following idea on which the proof is based. 4. A choice design of order n is a system of representatives of the triples of K: such that: (i) each point is chosen equally often as a representative; (ii) among the n - 2 triples containing a given pair {a, b}, a is chosen i ( n - 2 ) times and b i ( n - 2 ) times also. For example, when n =5, we have underlined the element chosen.

Thus R U ( B - A ) U D , c M and, by Lemma 5, ( RU (B - A ) U D o l s k. 17 Let now p=max{t: deg, ( w k + , ) a m - 3 k and G[wl, w2,.. , wk+t] contains 2t independent edges}. Lemma 6 implies that p 3 0 . Since 2p independent edges have 4p vertices, we have 4p s k + p so 0 s p s i k . Put W, = {wl,w2, . . , wktp). Lemma 7 . There is an x 1 - x 2 path P of length 2k in G I such that W,c V ( P )and X l Y l , X,YZEE(G) for Some Y 1 , Y 2 E A Y 1 + Y 2 . Proof. Choose a set S of 2 p independent edges in G[W,,].

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

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