In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta \ge 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta =7$ is edge $\Delta$-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every $K_5$-minor free graph with maximum degree $\Delta$ at least seven is edge $\Delta$-colorable.

中文翻译：

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的边缘着色 ${ķ}_5$-次要免费图表

1965年，Vizing证明了每个平面图 $G$ 最高学历 $\Delta \ge 8$ 是边缘 $\Delta$-彩色的。也证明了每个平面图$G$ 最高学历 $\Delta =7$ 是边缘 $\Delta$-由Sanders和Zhao着色，由Zhang独立着色。在本文中，我们通过展示每个${ķ}_5$-具有最大程度的次要免费图 $\Delta$ 至少有七个是优势 $\Delta$-彩色的。