It is a well-known conjecture that if a regular graph G of order 2n has degree d(G) satisfying d(G) ^ n, then G is the union of edge-disjoint 1-factors. It is well known that this conjecture is trueâ€¦ (More)

A star-multigraph G is a multigraph in which there is a vertex v* which is incident with each non-simple edge. It is critical if it is connected, Class 2 and x'(G\e) < ~'(G) for each e ~ E(G). Weâ€¦ (More)

In this note we prove the following theorem. THEOREM 1. Let k, t #s 1. Let s/ t = {A n , A t2 A lrt } be sets of subsets of {1, ..., m}, where U U r t), A hh * 0 (l < 'i < h < U l < h < r Â«i (C)' ifâ€¦ (More)

Philip Hallâ€™s famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of properâ€¦ (More)

Hilton, A. J. W., Alternating Hamiltonian circuits in edge-coloured bipartite graphs, Discrete Applied Mathematics 35 (1992) 271-273. We show that if G = K,, is edge-coloured with t-12 colours soâ€¦ (More)