Strong Edge Colouring of Disk Graphs: A 6-Approximation and an Improved Unit-Disk Bound
A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $χ'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chromatic index $χ'_{s,\ell}(G)$. From the definitions, $χ'_s(G)\leχ'_{s,\ell}(G)$. Barrett et al. gave an $8$-approximation for strong edge colouring on unit disk graphs and Grelier et al. improved the approximation factor to $6$. Our first result extends this factor-$6$ guarantee from unit disk graphs to the strictly larger class of disk graphs. In another direction, Erdős and Nešetřil conjectured that the strong chromatic index of a graph of maximum degree $Δ$ is asymptotically at most $1.25Δ^2$. The best published general asymptotic upper bound has leading coefficient $1.772$, due to Hurley et al. For unit disk graphs, Dębski et al. proved that $χ'_s(G) \leq 1.625 Δ^2$. Our second result improves this leading coefficient to $225/142 \approx 1.5845$. In fact, the proof establishes a stronger bound $χ'_{s,\ell}(G)\le\frac{225}{142} Δ^2+O(Δ)$ for unit disk graphs.