### On paths, trails and closed trails in edge-colored graphs

*Laurent Gourvès, Adria Ramos de Lyra, Carlos Alberto Martinhon, Jérôme Monnot*

#### Abstract

In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph Gc, we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in Gc, we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint pec paths (resp., trails) in Gc with endpoints in S. Further, if
Gc contains no pec closed trails, we show that the problem of finding a pec s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no pec cycles.
We also deal with graphs Gc containing no (almost) pec cycles or closed trails through s or t. We prove that finding 2 pec s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint pec s-t paths (resp., trails) so that the maximum path (resp., trail) length is
no more than k times the pec path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one pec s-t path is NPcomplete.
This result is interesting since as proved in [1], the determination of two or more vertex disjoint pec s-t paths can be done in polynomial time. Finally, if Gc is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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