Strong edge-colouring via local flag algebras

2026-07-19

AI Summary

This study addresses the strong edge-coloring problem, which requires assigning colors to edges of a graph such that any two edges at distance at most two receive distinct colors, with the goal of minimizing the number of colors used. The authors introduce, for the first time, the method of local flag algebras to this problem, combining it with probabilistic and extremal graph theory techniques to derive improved upper bounds on the strong chromatic index. Specifically, they establish bounds of $1.73\Delta^2$ for general graphs, $1.6255\Delta^2$ for bipartite graphs, and $1.6633\Delta_A\Delta_B$ for asymmetric bipartite graphs. These results improve upon the best-known bounds in the literature and nearly attain the Brualdi–Quinn Massey conjectured bound asymptotically almost surely in random bipartite graphs, thereby advancing progress on three classical conjectures in the field.

Abstract

The strong chromatic index $χ'_s(G)$ 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 $χ'_s(G) \leq 1.73\,Δ(G)^2$ for every graph $G$ of maximum degree $Δ(G)$, $χ'_s(G) \leq 1.6255\,Δ(G)^2$ for every bipartite $G$, and $χ'_s(G) \leq 1.6633\,Δ_A(G)\,Δ_B(G)$ for every bipartite $G$ of side maximum degrees $Δ_A(G), Δ_B(G)$ with rational $Δ_B(G)/Δ_A(G) \in (0, 1]$, provided $Δ(G)$, $Δ_A(G)$, $Δ_B(G)$ are sufficiently large. These three bounds make progress towards three established conjectures: those of Erdős-Nešetřil (1985) for general graphs, Faudree-Gyárfás-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 $χ'_s(G) \leq Δ_A(G)\,Δ_B(G)$ asymptotically almost surely.

Problem

• strong edge-colouring
• strong chromatic index
• maximum degree
• bipartite graphs
• asymmetric bipartite setting

Innovation

• strong edge-colouring
• local flag algebras
• strong chromatic index
• bipartite graphs
• asymptotic bounds