A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $\chi_{s}'(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. Erd\H{o}s and Ne\v{s}et\v{r}il conjectured in 1985 that the upper bound of $\chi_{s}'(G)$ is $\frac{5}{4}\Delta^{2}$ when $\Delta$ is even and $\frac{1}{4}(5\Delta^{2}-2\Delta +1)$ when $\Delta$ is odd, where $\Delta$ is the maximum degree of $G$. The conjecture is proved right when $\Delta\leq3$. The best known upper bound for $\Delta=4$ is 22 due to Cranston previously. In this paper we extend the result of Cranston to list strong edge-coloring, that is to say, we prove that when $\Delta=4$ the upper bound of list strong chromatic index is 22.

中文翻译：

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列出最大度数为 4 的图形的强边缘着色

图 $G$ 的强边着色是一种边着色，使得长度为 3 的路径上的任何两条边都收到不同的颜色。我们用 $\chi_{s}'(G)$ 表示强色指数，这是允许 $G$ 强边缘着色的最小颜色数。Erd\H{o}s 和 Ne\v{s}et\v{r}il 在 1985 年推测 $\chi_{s}'(G)$ 的上限是 $\frac{5}{4} \Delta^{2}$ 当 $\Delta$ 为偶数时 $\frac{1}{4}(5\Delta^{2}-2\Delta +1)$ 当 $\Delta$ 为奇数时，其中 $ \Delta$ 是$G$ 的最大度数。当 $\Delta\leq3$ 时，猜想被证明是正确的。由于先前的克兰斯顿，$\Delta=4$ 最著名的上限是 22。在本文中，我们将 Cranston 的结果扩展到列表强边着色，也就是说，我们证明了当 $\Delta=4$ 时，列表强色指数的上限为 22。