Cycles of given color patterns

Abstract

In 2-edge-colored graphs, we define an (s, t)-cycle to be a cyle of length s + t, in which s consecutive edges are in one color and the remaining t edges are in the other color. Here we investigate the existence of (s, t)-cycles, in a 2-edge-colored complete graph Kcn on n vertices. In particular, in the first result we give a complete characterization for the existence of (s, t)-cycles in Kcn with n relatively large with respect to max({s, t}). We also study cycles of length 4 for all possible values of s and t. Then, we show that Kcn contains an (s, t)-hamiltonian cycle unless it is isomorphic to a specified graph. This extends a result of A. Gyárfás [Journal of Graph Theory, 7 (1983), 131–135]. Finally, we give some sufficient conditions for the existence of (s, 1)-cycles, (inverted sans serif aye) s ϵ {2, 3,…, n − 2}. © 1996 John Wiley & Sons, Inc.