Abstract For a vertex v of an edge-colored graph, the color degree of v is the number of colors appeared in edges incident with v . An edge-colored graph is called properly colored if no two adjacent edges have the same color. In this paper, we prove that if the minimum color degree sum of two adjacent vertices of an edge-colored connected graph G is at least | G | , then G has a properly colored spanning tree. This is a generalization of the result proved by Cheng, Kano and Wang. We also show the sharpness of this lower bound of the color degree sum.

中文翻译：

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边着色图中正确着色生成树的颜色度和条件

摘要 对于边着色图的顶点 v，v 的颜色度是与 v 相关的边中出现的颜色数。如果没有两条相邻的边具有相同的颜色，则称边着色图正确着色。在本文中，我们证明如果边色连通图G的两个相邻顶点的最小色度和至少为| G | ，那么 G 有一个正确着色的生成树。这是对Cheng、Kano 和Wang 证明的结果的概括。我们还展示了色度总和的下限的锐度。