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

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Throughout the paper we use the terminology and notation of [ l ] . Theorem 1. Let k and m be natural numbers satisfying m 2 9 k . Let G be a , Then G is Hamil2-connected ( m - k)-regular graph of order 2m - E , E ~ ( 0I}. tonian. Proof. In order to reduce the number of symbols floating around, we shall take E = 0. The case E = 1 can be treated in exactly the same way. 43 44 B. Bollobas, A. Hobbs Fig. 1. A two-connected 4-regular non-Hamiltonian graph of order 16, k = 4. rn =8 Let us assume that G does not contain a Hamiltonian cycle.

Exactly as in the proof of Theorem 7, we can prove that G " contains at least c , n J cycles CJ,where c , > O is a constant. Applying the theorem of Erdos on hypergraphs [4], we obtain j sets X I , .. , X I with IXll= T+m, such that if x , E X , , . . , . ,x ~ , )is a cycle of G" (we consider here the hypergraph whose hyperedges are the j-sets of vertices of j-cycles in G"). Unfortunately the cycles will not determine a CJ(T), since the permutation i,. . , i, may differ from j-tuple to j-tuple.

N } we choose the element defined by the choice design of order n. For the triples we choose (i, j , a ) (i,j , P ) (i, j , Y) with i < j , i y if i + j = O (mod3), j P i if i + j = l (mod3), a i j if i + j = 2 (mod3). For the triples we choose (i, a, S) (i, a, Y) (2, P, Y) 1 Y y P a 1 a i P if i E 0 (mod 3), if i = 1 (mod 3), if i = 2 (mod3). For the triple ( a , p, y ) we choose y. We leave to the reader the care of checking that we obtain a choice design of order n + 3; the only non-immediate part is to check property (ii) for the triples containing a pair (i, a ) or (i, p ) or (i, y ) .

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

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