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.

H , be an enumeration of the right cosets of S,, with respect to H. Let 51 Chromatic index of the graph of the assignment polytope G , ( H i ) denote the subgraph of G, induced by the vertices in Hi(1s i c t). For distinct integers i and j with 1 s i, j c t, let Gn(Hi,Hi) denote the partial subgraph of G,, where the set of vertices is Hi U Hi and where two vertices are joined by an edge if and only if one is in Hi, the other is in Hi, and they are joined by an edge in G,. In particular Gn(Hi,Hi) is a bipartite graph with bipartition Hi, Hi.

Then each a, ( i = 1 , 2 ) is joined to at most p - 1 vertices of R and so at least 2 ( m - k - p + 1) edges join { u l ,a2} to L. I f x1 and x2 are vertices of L at distance d on L and 1c d =sp , then either Y , is not adjacent to a , or x2 is not adjacent to a2, since otherwise there is a cycle longer than L, as shown in Fig. 3 . Hence at most two edges join any set of p + 1 consecutive vertices of L to the set { a , , a2}. Consequently at most 2/(p+ 1)x (2m - r ) edges join { a , , a,} to L so 2 ( m - k - p + 1)< 2 ( 2 m - r ) / ( p + 1).

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

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