Edge-disjoint Hamilton Cycles in Regular Graphs of Large Degree

Abstract

Theorem 1 implies that if G is a A:-regular graph on n vertices and n ^ 2k, then G contains Wr(n + 3an + 2)] edge-disjoint Hamilton cycles. Thus we are able to increase the bound on the number of edge-disjoint Hamilton cycles by adding a regularity condition. In [5] Nash-Williams conjectured that if G satisfies the conditions of Theorem 2, then G contains [i(n + l)] edge-disjoint Hamilton cycles. Although counterexamples were subsequently constructed by L. Babai, NashWilliams points out that his conjecture remains open for regular graphs [6; pp. 817-818]. We conjecture further, that if G is a A>regular graph on n vertices and n^2k+\ then G contains [%k] edge-disjoint Hamilton cycles. This conjecture is similar to one due to P. Kelly [4; p. 7] that any regular tournament can be decomposed into directed Hamilton cycles, since both conjectures imply that K2m+l is, in some sense, "strongly decomposable" into Hamilton cycles. Our conjecture would imply that any set of [i(m-l-l)] edge-disjoint Hamilton cycles of K2m+i could be extended to a hamiltonian decomposition of K2m+l. Kelly's conjecture would imply that if the edges of K2m+1 are directed in such a way that the indegrees and outdegrees of every vertex are equal, then the resulting digraph has a hamiltonian decomposition. For any graph G, let V{G) denote the set of vertices, and E(G) the set of edges, of G. For H a subgraph of G and M and N subsets of V(H) let EH{M, N) denote the set, and ew(M, N) the number of edges between the vertices of M and the vertices of N. Further, let H [M] be the subgraph induced in H by the vertices of M. For veV(H) put dH(v) = sH({v}, V(H)\{v}). In the following, G will always denote a graph on n vertices. In order to simplify notation we shall write V, E(M, N), e(M, N), and d(v) for V(G), EG(M, AT), eG(M, N) and dG(v) respectively. We shall use the following two results. The first is due to Erdos and Gallai [2; p. 344] while the second is due to Nash-Williams [5; Lemma 7]. Both follow easily from Chvatal's Theorem on Hamilton cycles [1; Theorem 1].

Cite this paper

@inproceedings{Jackson1979EdgedisjointHC, title={Edge-disjoint Hamilton Cycles in Regular Graphs of Large Degree}, author={Bill Jackson}, year={1979} }