Maximum properly colored trees in edge-colored graphs

n edge-colored graph G is a graph with an edge coloring. We say G is properly colored if any two adjacent edges of G have distinct colors, and G is rainbow if any two edges of G have distinct colors. For a vertex v ∈ V(G) , the color degree d_G^col(v) of v is the number of distinct colors appearing on edges incident with v . The minimum color degree δ ^col(G) of G is the minimum d_G^col(v) over all vertices v ∈ V(G) . In this paper, we study the relation between the order of maximum properly colored tree in G and the minimum color degree δ ^col(G) of G . We obtain that for an edge-colored connected graph G , the order of maximum properly colored tree is at least min{|G|, 2δ ^col(G)} , which generalizes the result of Cheng et al. [Properly colored spanning trees in edge-colored graphs, Discrete Math., 343 (1), 2020]. Moreover, the lower bound 2δ ^col(G) in our result is sharp and we characterize all extremal graphs G with the maximum properly colored tree of order 2δ ^col(G) |G| .