Properly colored cycles in edge-colored complete graphs without monochromatic triangle: A vertex-pancyclic analogous result

Abstract

A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is known that every edge-colored complete graph without monochromatic triangle always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the exceptions. As a consequence, we partially confirm a structural conjecture given by Bollobás and Erdős (1976) and an algorithmic conjecture given by Gutin and Kim (2009).