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

Abstract

In an edge-colored graph (G,c), let dc(v) denote the number of colors on the edges incident with a vertex v of G and δc(G) denote 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. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that (G,c) does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.