Rainbow and Properly Colored Spanning Trees in Edge-Colored Bipartite Graphs

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.