通过局部标志代数进行强边着色
Strong edge-colouring via local flag algebras
AI总结：
研究通过局部标志代数对图的强边着色问题，给出了一般图、二分图及特定二分图的强色指数上界，朝着相关猜想取得进展，还证明了随机二分图在特定条件下的布鲁阿尔迪 - 奎因·梅西界。
AI中文摘要：
强色指数\(\chi'_s(G)\)是给图\(G\)的边着色所需的最小颜色数，使得距离至多为\(2\)的任意两条边接收不同颜色。利用在一篇相关论文中引入的“局部标志代数”框架，我们证明对于最大度为\(\Delta(G)\)的每个图\(G\)，\(\chi'_s(G) \leq 1.73\,\Delta(G)^2\)；对于每个二分图\(G\)，\(\chi'_s(G) \leq 1.6255\,\Delta(G)^2\)；对于边最大度为\(\Delta_A(G)\)、\(\Delta_B(G)\)且\(\Delta_B(G)/\Delta_A(G) \in (0, 1]\)为有理数的每个二分图\(G\)，当\(\Delta(G)\)、\(\Delta_A(G)\)、\(\Delta_B(G)\)足够大时，\(\chi'_s(G) \leq 1.6633\,\Delta_A(G)\,\Delta_B(G)\)。这些三个界朝着三个既定猜想取得了进展。此外，对于随机二分图\(G \sim G(n_A, n_B, p)\)，在常数\(p \in (0,1)\)且有界纵横比\(\max(n_A, n_B) = O(\min(n_A, n_B))\)的情况下，我们渐近几乎必然地证明了布鲁阿尔迪 - 奎因·梅西界\(\chi'_s(G) \leq \Delta_A(G)\,\Delta_B(G)\)。
英文摘要：
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.
