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

By B. Bollobás (Eds.)

ISBN-10: 0720408431

ISBN-13: 9780720408430

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Hung. 22 (3-4) (1971) 275-282. [8] P. H. Stone, On the structure of linear graphs, Bull. Amer. Math. 52 (1946) 1087-1091. [9] A. RCnyi, Probability Theory (North-Holland, Amsterdam, 1970). [lo] M. Simonovits, A method for solving extremal problems in graph theory, stability problems, in: Theory of Graphs, Proc. Coll. Tihany, Hungary (1966) 279-319. [ 1 11 E. Szemeredi, Regular partitions of graphs (to appear). This Page Intentionally Left Blank Annals of Discrete Mathematics 3 (1978) 43-48. @ North-Holland Publishing Company HAMILTONIAN CYCLES IN REGULAR GRAPHS Btla BOLLOBAS Department of Pure Mathematics and Mathematical Statistics, Cambridge CB2 1SB, England Arthur HOBBS Texas A.

Unfortunately the cycles will not determine a CJ(T), since the permutation i,. . , i, may differ from j-tuple to j-tuple. However, let us 40 B. Bollobas, P. Erdos, M. Simonovits, E. Szemeridi apply the Erdos theorem again, now to the hypergraph whose vertices are in X , U X,U. . UX, and the hyperedges of which are some cycles of G" of form ( x , ~ ,. . , x,,),x, E X,, where we choose only one permutation i,, . . If. , . , x,)defines a cycle in G" and I Y,I = t. Thus we obtained a C 1 ( tc) G".

Conjecture. One can easily ask many other problems, for example the directed versions of the results or problems of Section 1. But there are also problems peculiar to the directed case like Kelly’s conjecture that every regular tournament can be decomposed into directed hamiltonian cycles. 3. 1. For hypergraphs the number of problems grows quickly, because one can give different definitions of a hamiltonian cycle. I will restrict myself to a definition and a problem considered in [7]. 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 , .

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


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