Properly colored cycles in edge-colored 2-colored-triangle-free complete graphs

Abstract

An edge-colored graph Gc is called properly colored if any two adjacent edges receive distinct colors. An edge-colored graph Gc is 2-colored-triangle-free if Gc contains no 2-colored-triangle, where a 2-colored-triangle is an edge-colored triangle with exactly two colors. Let dc(v) be the number of colors on the edges incident to v in Gc and let δc(Gc) be the minimum dc(v) for all v∈V(Gc). In this paper we extend the definitions of proper vertex-pancyclic and proper edge-pancyclic to proper k-path-pancyclic, defined as follows: An edge-colored graph Gc is said to be proper k-path-pancyclic if each properly colored path of length k is contained in a properly colored cycle of length l for every l with max{3,k+2}≤l≤|V(Gc)|. We prove that an edge-colored 2-colored-triangle-free complete graph Gc with δc(Gc)≥k+3 is either (almost) proper k-path-pancyclic or contains a large monochromatic complete subgraph.