Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic paths of length three

Abstract

Let Gc be an edge-colored graph of order n. The minimum color degree δc(Gc) of Gc is the minimum number of different colors appearing on the edges incident with a vertex of Gc. A cycle in Gc is proper if no two adjacent edges are assigned the same color. The graph Gc is called properly vertex-pancyclic if each vertex of Gc is contained in a proper cycle of length k for every k with 3≤k≤n. We show in this paper that if δc(Knc)≥n+12 and Knc contains no monochromatic P4, then Knc is properly vertex-pancyclic. Our result partially solves a conjecture proposed by Fujita and Magnant which says that, if δc(Knc)≥n+12, then Knc is properly vertex-pancyclic.