Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic triangles

Abstract

In an edge-colored graph (G,c), let dc(v) be the number of colors on the edges incident to vertex v and let δc(G) be the minimum value of dc(v) over all vertices v∈V(G). A cycle of (G,c) is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph (G,c) on n≥3 vertices is called properly vertex-pancyclic if each vertex of (G,c) is contained in a proper cycle of length l for every l with 3≤l≤n. Fujita and Magnant conjectured that every edge-colored complete graph on n≥3 vertices with δc(G)≥n+12 is properly vertex-pancyclic. We show that this conjecture is true if the edge-colored complete graph has no monochromatic triangles.