Monochromatic-degree conditions for properly colored cycles in edge-colored complete graphs

Abstract

Let G be an edge-colored graph and v be a vertex of G. Define the monochromatic-degree dmon(v) of v to be the maximum number of edges with the same color incident with v in G, and the maximum monochromatic-degree Δmon(G) of G to be the maximum value of dmon(v) over all vertices v of G. A cycle (path) in G is called properly colored if any two adjacent edges of the cycle (path) have distinct colors. Wang et al. in 2014 showed that an edge-colored complete graph Knc with Δmon(Knc)<⌊n2⌋ contains a properly colored cycle of length at least ⌈n2⌉+2. In this paper, we obtain a generalization of their result that an edge-colored complete graph Knc of order n with Δmon(Knc)=d≤n−2 contains a properly colored cycle of length at least n−d+1.