Long alternating cycles in edge-colored complete graphs

Abstract

Let Knc denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let Δ(Knc) denote the maximum number of edges of the same color incident with a vertex of Knc. A properly colored cycle (path) in Knc, that is, a cycle (path) in which adjacent edges have distinct colors is called an alternating cycle (path). Our note is inspired by the following conjecture by B. bollobás and P. Erdos(1976): If Δ(Knc)