Complexity separating classes for edge-colouring and total-colouring

Raphael Machado,Celina De Figueiredo

doi:10.1007/s13173-011-0040-8

Abstract

Abstract The class of unichord-free graphs was recently investigated in a series of papers (Machado et al. in Theor. Comput. Sci. 411:1221–1234, 2010; Machado, de Figueiredo in Discrete Appl. Math. 159:1851–1864, 2011; Trotignon, Vušković in J. Graph Theory 63:31–67, 2010) and proved to be useful with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies could be found in subclasses of unichord-free graphs. We discuss such results based on the concept of “separating class” and we describe the class of bipartite unichord-free as a final missing separating class with respect to edge-colouring and total-colouring problems, by proving that total-colouring bipartite unichord-free graphs is NP-complete.

Highlights

Given a class G of graphs and a graph problem φ belonging to NP, we say that a full complexity dichotomy of G was obtained if one describes a partition of G into G1, G2, . . . such that φ is classified as polynomial or NPcomplete when restricted to each Gi
The concept of full complexity dichotomy is interesting for the investigation of NP-complete problems: as we partition a class
If a problem φ is polynomial in G, any partition of G will determine polynomial subclasses; and if a problem is NP-complete in G, any finite partition G1, G2, . . . , Gn of G will determine at least one subclass Gi such that φ restricted to Gi is NP-complete or the recognition of Gi is NP-complete

Summary

Introduction

Given a class G of graphs and a graph (decision) problem φ belonging to NP, we say that a full complexity dichotomy of G was obtained if one describes a partition of G into G1, G2, . . . such that φ is classified as polynomial or NPcomplete when restricted to each Gi. Recall that both edge-colouring and total-colouring are NP-complete problems and consider the classes We invite the reader to check that each Gx with even x is a separating class where edge-colouring is polynomial and total-colouring is NP-complete.

Results

Conclusion