Odd Properly Colored Cycles in Edge-Colored Graphs

TL;DR

This paper investigates the existence of odd properly colored cycles in edge-colored graphs, generalizing bipartiteness concepts from undirected and directed graphs to more complex edge-colored structures.

Contribution

It introduces methods to determine the presence of odd properly colored cycles and relates this to detecting perfect matchings in edge-colored graphs.

Findings

Decidable criteria for odd properly colored cycles

Connection between odd cycles and perfect matchings

Extension of bipartiteness concepts to edge-colored graphs

Abstract

It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this result be further generalized to more general graphs such as edge-colored graphs? In this paper, we study this problem and show how to decide if there exists an odd properly colored cycle in a given edge-colored graph. As a by-product, we show how to detect if there is a perfect matching in a graph with even (or odd) number of edges in a given edge set.