Download PDF by B. Bollobás (Eds.): Advances in Graph Theory

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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If H is a hypergraph with n vertices then a hamiltonian cycle is a sequence x , E l x 2 * * x,EIxI+, x,E,,x, such that Hamilronian decompositions of graphs, directed graphs and hypergraphs 25 (i) the n vertices xi are all different (and thus are the n vertices of the hypergraph) > (ii) the n edges Eiare all different, ~ n -{ xl ,), x , ) c E , . (iii) { x ~ , x i * ~ } ~ E i ( l ~ i and Let Kf: denote the complete h-uniform hypergraph; its edges are all the h-subsets of a set X of cardinality n.

X. Bermond and V. Faber, in: Problems of the 5th British Combinatorial Conference, Aberdeen 1975, Utilitas Math. Congressus Numerantium XV (1975). 695. -C. Bermond and V. Faber, Decomposition of the complete directed graph into k-circuits, J . Cornbinatorial Theory, 21(B) (1976) 146-155. -C. Bermond and D. Sotteau, Graph Decompositions and G-designs, Proc. 5th Combinatorial Conference, Aberdeen 1975, Utilitas Math. -C. Bermond, A. -C. Heydemann and D. Sotteau, Hypergraphes hamiltoniens, in: Coll.

Put M = { wi : deg, ( wi)3 m - 3 k } . Lemma 6. [MI3 k, that is deg, (wk) 2 m - 3k. Proof. By Lemma 2 every vertex of R has degree at least m - k - 1 in W. Furthermore no vertex of (B - A ) U Do is joined to any vertex of A f l B. Hence each vertex of ( B - A ) U D , has degree at least m - k - s 3 m - k - ( 2 k - r ) ~ m-3k+r in W. 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}.

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

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