An edge-colored graph is called rainbow (or heterochromatic) if all its edges have distinct colors. It is known that if an edge-colored connected graph H has minimum color degree at least |H|/2 and has a certain property, then H has a rainbow spanning tree. In this paper, we prove that if an edge-colored connected bipartite graph G has minimum color degree at least |G|/3 and has a certain property, then G has a rainbow spanning tree. We also give a similar sufficient condition for G to have a properly colored spanning tree. Moreover, we show that these minimum color-degree conditions are sharp.

中文翻译：

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边着色二部图中的彩虹和正确着色的生成树

如果一个边色图的所有边都具有不同的颜色，则它被称为彩虹（或异色）。已知如果一个边有色连通图H至少有最小色度| H |/2 且具有一定的性质，则H有彩虹生成树。在本文中，我们证明如果一个边着色连通二部图G至少有最小色度 | G |/3 且具有一定的性质，则G有彩虹生成树。我们还给出了G具有适当着色的生成树的类似充分条件。此外，我们表明这些最小色度条件是清晰的。