Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

Zhiwei Guo,Shenggui Zhang,Hajo Broersma,Ruonan Li

doi:10.1007/s10878-020-00644-7

Abstract

A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

Highlights

In this paper we consider only finite undirected simple graphs
The ideas behind Algorithm 1 were inspired by similar ideas due to Pevzner (1995) for an efficient algorithm to construct a compatible Euler tour in an edge-colored eulerian graph G in which mon(v) ≤ d(v)/2 for each vertex v of G
We considered the existence of more general compatible spanning circuits in edge-colored graphs from an algorithmic perspective

Summary

Introduction

In this paper we consider only finite undirected simple graphs. For terminology and notations not defined here, we refer the reader to the textbook of Bondy and Murty (2008).Let G be a graph. Keywords Edge-colored graph · Compatible spanning circuit · NP-complete problem · Polynomial-time algorithm Benkouar et al (1996) provided a polynomial-time algorithm for finding a compatible Euler tour in an edge-colored eulerian graph G in which mon(v) ≤ d(v)/2 for each vertex v of G.

Results

Conclusion