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

An edge-colored graph G c is called properly colored if any two adjacent edges receive distinct colors. An edge-colored graph G c is 2-colored-triangle-free if G c contains no 2-colored-triangle, where a 2-colored-triangle is an edge-colored triangle with exactly two colors. Let d c (v) be the number of colors on the edges incident to v in G c and let δ c (G c) be the minimum d c (v) for all v ∈ V (G c). 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 G c 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 (G c) |. We prove that an edge-colored 2-colored-triangle-free complete graph G c with δ c (G c) ≥ k + 3 is either (almost) proper k -path-pancyclic or contains a large monochromatic complete subgraph. [ABSTRACT FROM AUTHOR]