Properly colored cycles in edge-colored complete graphs containing no monochromatic triangles: a vertex-pancyclic analogous result

TL;DR

This paper investigates the existence of properly colored cycles in complete graphs with no monochromatic triangles, establishing a vertex-pancyclic-like result and characterizing exceptions.

Contribution

It extends known results by proving a vertex-pancyclic analogous theorem for such graphs and identifying all exceptions.

Findings

Existence of properly colored cycles in graphs without monochromatic triangles.

Characterization of all exceptions to the main result.

Extension of Hamilton path results to cycles.

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 containing monochromatic triangles 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.