Abstrakt
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.