Properly Coloured Cycles and Paths: Results and Open Problems

TL;DR

This paper reviews known results, introduces new findings, and discusses open problems related to properly edge-coloured paths and cycles in multigraphs, including transformations to ordinary graphs for detection and optimization.

Contribution

It presents new methods for transforming edge-coloured multigraphs into ordinary graphs to identify and optimize properly coloured cycles and paths, and discusses open conjectures.

Findings

New transformations for detecting PC cycles and paths

Existence and shortest PC cycle/path algorithms

Open problems and conjectures in PC graph theory

Abstract

In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of transformations of an edge-coloured multigraph $G$ into an ordinary graph that allow us to check the existence PC cycles and PC $(s,t)$-paths in $G$ and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible solution to the problem.