Properly Coloured Cycles and Paths: Results and Open Problems

Abstract

In this paper, we consider a number of results and six conjectures on properly 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 of 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.

Long properly colored cycles in edge colored complete graphs

“…Li, Wang and Zhou [12] studied long properly colored cycles in edge colored complete graphs and proved that if ∆ mon (K c n ) < n 2 , then K c n contains a properly colored cycle of length at least n+2 3 + 1. For more details concerning properly colored cycles and paths, we refer the reader to [2,10,11,16]. In this paper, we improve the bound on the length of the properly colored cycles and prove the following theorem.…”

Complexity of Paths, Trails and Circuits in Arc-Colored Digraphs

“…Problems regarding properly edge-colored paths, trails and cycles (or pec paths, trails and cycles, for short) in c-edge-colored (undirected) graphs have been widely studied from a graph theory and algorithmic point of views (see [3,1,24], the book [5] and the recent survey [18]). For instance, in [23], the author gives polynomial algorithms for several problems, including the determination of a pec s-t path (if one exists).…”

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

“…Bollobás and Erdős [5] initiated the study of properly edge-colored Hamiltonian cycles (which they called alternating Hamiltonian cycles) in edge-colored complete graphs, and this study was continued in papers such as [1,2]. The problem of finding properly edge-colored paths and circuits in edge-colored digraphs has been studied in several articles such as Gourvès et al [13]; see the survey paper by Gutin and Kim [14] for an overview. Finding subgraphs with all edges having different colors (called rainbow or heterochromatic) has also been well studied.…”