A strong edge coloring of a graph $G$ is an edge coloring $φ\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $χ'_s(G)$, is the minimum number of colors needed for a strong edge coloring of $G$. Erdős and Nešetřil conjectured in 1985 that if $G$ has maximum degree $d$, then $χ'_s(G) \leq \frac 54 d^2$. Mahdian showed in 2000 that if $G$ is $C_4$-free, then $χ'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$, and he conjectured that the same upper bound holds for $K_{t,t}$-free graphs. In this paper, we prove this conjecture and improve upon it to show the following: every $K_{t,t}$-free graph $G$ of maximum degree $d$ satisfies $χ'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$. We employ a variant of the Rödl nibble method to prove this result. The key new ingredient in our adaptation of the method is an application of the Kővári-Sós-Turán theorem to show that $H := L(G)^2$ satisfies certain structural properties. These properties, in conjunction with a variant of Talagrand's inequality to handle exceptional outcomes, allow us to concentrate the sizes of certain vertex sets through the nibble, even when these vertex sets have order smaller than the maximum codegree of $H$. We encapsulate these structural properties into a more general statement on list coloring that we believe to be of independent interest. In light of the conjectured computational threshold for coloring random graphs arising in average-case complexity theory, we suspect that our result is best possible using this approach.

翻译：图 $G$ 的强边染色是指一种边染色 $\varphi\,:\,E(G) \rightarrow \mathbb N$，使得每个颜色类在 $G$ 中构成一个诱导匹配。$G$ 的强色指数，记作 $\chi'_s(G)$，是为 $G$ 进行强边染色所需的最少颜色数。Erdős 和 Nešetřil 于 1985 年猜想：若 $G$ 的最大度为 $d$，则 $\chi'_s(G) \leq \frac 54 d^2$。Mahdian 于 2000 年证明了若 $G$ 为 $C_4$-自由图，则 $\chi'_s(G) \leq (2+o(1)) \frac{d^2}{\log d}$，并猜想相同上界对 $K_{t,t}$-自由图也成立。本文证明并改进该猜想，得到如下结论：每个最大度为 $d$ 的 $K_{t,t}$-自由图 $G$ 满足 $\chi'_s(G) \leq (1+o(1)) \frac{d^2}{\log d}$。我们采用 Rődl nibble 方法的一种变体来证明此结果。该方法改编的关键新要素是应用 Kővári-Sós-Turán 定理证明 $H := L(G)^2$ 满足特定结构性质。这些性质结合 Talagrand 不等式（用于处理异常情况）的变体，使我们能够在 nibble 过程中集中某些顶点集的大小，即使这些顶点集的阶数小于 $H$ 的最大余度。我们将这些结构性质封装为关于列表染色的更一般性陈述，相信该陈述具有独立研究价值。考虑到平均情况复杂度理论中随机图着色的计算阈值猜想，我们认为本结果在此方法下已达到最优。