Strong edge-colouring via local flag algebras

The strong chromatic index $\chi'_s(G)$ is the smallest number of colours needed to colour the edges of a graph $G$ so that any two edges at distance at most $2$ receive different colours. Using the \emph{local flag algebra} framework introduced in a companion paper, we prove $\chi'_s(G) \leq 1.73\,\Delta(G)^2$ for every graph $G$ of maximum degree $\Delta(G)$, $\chi'_s(G) \leq 1.6255\,\Delta(G)^2$ for every bipartite $G$, and $\chi'_s(G) \leq 1.6633\,\Delta_A(G)\,\Delta_B(G)$ for every bipartite $G$ of side maximum degrees $\Delta_A(G), \Delta_B(G)$ with rational $\Delta_B(G)/\Delta_A(G) \in (0, 1]$, provided $\Delta(G)$, $\Delta_A(G)$, $\Delta_B(G)$ are sufficiently large. These three bounds make progress towards three established conjectures: those of Erd\H{o}s-Ne\v{s}et\v{r}il (1985) for general graphs, Faudree-Gy\'arf\'as-Schelp-Tuza (1989) for bipartite graphs, and Brualdi-Quinn Massey (1993) in the asymmetric bipartite setting. Additionally, for the random bipartite graph $G \sim G(n_A, n_B, p)$ at constant $p \in (0,1)$ and bounded aspect ratio $\max(n_A, n_B) = O(\min(n_A, n_B))$, we prove the Brualdi-Quinn Massey bound $\chi'_s(G) \leq \Delta_A(G)\,\Delta_B(G)$ asymptotically almost surely.