We develop sequential algorithms for constructing edge-colorings of graphs and multigraphs efficiently and using few colors. Our primary focus is edge-coloring arbitrary simple graphs using $d+1$ colors, where $d$ is the largest vertex degree in the graph. Vizing's Theorem states that every simple graph can be edge-colored using $d+1$ colors. Although some graphs can be edge-colored using only $d$ colors, it is NP-hard to recognize graphs of this type [Holyer, 1981]. So using $d+1$ colors is a natural goal. Efficient techniques for $(d+1)$-edge-coloring were developed by Gabow, Nishizeki, Kariv, Leven, and Terada in 1985, and independently by Arjomandi in 1982, leading to algorithms that run in $O(|E| \sqrt{|V| \log |V|})$ time. They have remained the fastest known algorithms for this task. We improve the runtime to $O(|E| \sqrt{|V|})$ with a small modification and careful analysis. We then develop a randomized version of the algorithm that is much simpler to implement and has the same asymptotic runtime, with very high probability. On the way to these results, we give a simple algorithm for $(2d-1)$-edge-coloring of multigraphs that runs in $O(|E|\log d)$ time. Underlying these algorithms is a general edge-coloring strategy which may lend itself to further applications.
翻译：我们开发了用于构建图表和多面图的边色和边色的序列算法, 并且使用了很少的颜色。 我们的主要焦点是使用 $d+1 彩色的边色任意的简单图表, 使用 $d+1 彩色, 美元是图形中最大的顶点度。 Vizing 的Theorem 表示, 每个简单的图表都可以使用 $+1 彩色 。 虽然有些图表只能使用 $d 彩色, 却很难识别这种类型的图表 [ 1981 。 因此, 使用 $+1 彩色是一个自然目标。 1985年, Gabow、 Nishizeki、 Kariv、 Leven 和 Terada 开发了 高亮度技术。 1982 由 Arjomandi 独立开发了每个简单的彩色, 导致以 $( ⁇ ⁇ \\\\\\\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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