Properly colored spanning trees in edge-colored graphs

A subgraph H of an edge-colored graph G is called a properly colored subgraph if no two adjacent edges of H have the same color, and is called a rainbow subgraph if no two edges of H have the same color. For a vertex v of G, the color degree of v, denoted by dGc(v), is the number of distinct colors appeared in the edges incident with v. Let δc(G) be the minimum value among the color degrees of all the vertices in G. We prove the following two theorems and show that the conditions on the minimum color degree are sharp. Let G be an edge-colored graph. If δc(G)≥|G|∕2, then G has a properly colored spanning tree. Moreover, if δc(G)≥|G|∕2 and the set of edges colored with any fixed color forms a subgraph of order at most (|G|∕2)+1, then G has a rainbow spanning tree.