Acyclicity in edge-colored graphs

Abstract

A walk W in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in W is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type i is a proper superset of graphs of acyclicity of type i + 1, i = 1, 2, 3, 4. The first three types are equivalent to the absence of PC cycles, PC closed trails, and PC closed walks, respectively. While graphs of types 1, 2 and 3 can be recognized in polynomial time, the problem of recognizing graphs of type 4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when only two colors are used). The same problem with respect to type 5 is polynomial-time solvable for all edge-colored graphs. Using the five types, we investigate the border between intractability and tractability for the problems of finding the maximum number of internally vertex-disjoint PC paths between two vertices and the minimum number of vertices to meet all PC paths between two vertices.

Odd properly colored cycles in edge-colored graphs

“…A natural question is whether this problem can be solved in (deterministic) polynomial time. It was proved in [6] that if an edge-colored graph G has no PC closed walk then G has a monochromatic vertex. This can be viewed as a characterization of edge-colored graphs with no PC closed walk and it implies that deciding whether G has a PC closed walk is polynomial-time solvable.…”

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

“…Furthermore, the bound is best possible. Besides, Yeo's theorem is frequently used not only on the existence of PC cycles (see the survey paper [1] and recent results [8,13,17,19]) but also on other topics, such as PC trees [7], decomposition of edge-colored graphs [12],…”