Maximum edge colouring problem on graphs that exclude a fixed minor

TL;DR

This paper investigates the maximum edge colouring problem in sparse graphs, proving NP-hardness for 1-apex graphs and developing a PTAS for minor-free graphs using a novel Baker game approach.

Contribution

It introduces a PTAS for the problem on minor-free graphs and demonstrates NP-hardness on 1-apex graphs, expanding the applicability of the Baker game technique.

Findings

NP-hardness on 1-apex graphs

Existence of PTAS for minor-free graphs

Baker game technique extends beyond first-order logic

Abstract

The maximum edge colouring problem considers the maximum colour assignment to edges of a graph under the condition that every vertex has at most a fixed number of distinct coloured edges incident on it. If that fixed number is $q$ we call the colouring a maximum edge $q$-colouring. The problem models a non-overlapping frequency channel assignment question on wireless networks. The problem has also been studied from a purely combinatorial perspective in the graph theory literature. We study the question when the input graph is sparse. We show the problem remains $NP$-hard on $1$-apex graphs. We also show that there exists $PTAS$ for the problem on minor-free graphs. The $PTAS$ is based on a recently developed Baker game technique for proper minor-closed classes, thus avoiding the need to use any involved structural results. This further pushes the Baker game technique beyond the problems expressible in the first-order logic.