  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.
