Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge colored with at most $\Delta+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on $n$ vertices with maximum degree $\Delta = \omega(\log n)$ using at most $(1+o(1))\Delta$ colors. Here we study the na\"{i}ve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can $(1+\epsilon)\Delta$-color the graph for arbitrary $\epsilon$ in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., $n = O(\Delta)$. Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which $(1+\epsilon)\Delta$ edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized $2\Delta-1$ colors.

翻译：维津定理指出，任何最大度为$\Delta$的图最多可用$\Delta+1$种颜色实现正常边着色。在线设定中，寻找一种算法能够用最多$(1+o(1))\Delta$种颜色对任意$n$个顶点且最大度$\Delta = \omega(\log n)$的图进行正常边着色，一直是研究热点。本文研究朴素的随机贪婪算法——该算法仅在每条边到达时从合法颜色中均匀随机选择一种着色。我们证明该算法在两种情境下能以$(1+\epsilon)\Delta$种颜色对图进行着色（$\epsilon$为任意值）：首先，当边以均匀随机顺序到达时；其次，当边以对抗顺序到达但图足够稠密（即$n = O(\Delta)$）时。在本研究之前，随机贪婪算法仅已知在树结构中有效。我们的第二个结果甚至适用于自适应对抗情境，这暗示了存在一种确定性边着色算法，能以$(1+\epsilon)\Delta$种颜色对稠密图进行边着色。此前，该问题最著名的确定性算法是使用$2\Delta-1$种颜色的简单贪婪算法。