Maximum properly colored trees in edge-colored graphs

An 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 is an element of 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 delta(col) (G) of G is the minimum d(G)(col) (v) over all vertices v is an element of V(G). In this paper, we study the relation between the order of maximum properly colored tree in G and the minimum color degree delta(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{vertical bar G vertical bar, 2 delta(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 2 delta(col) (G) in our result is sharp and we characterize all extremal graphs G with the maximum properly colored tree of order 2 delta(col) (G) not equal vertical bar G vertical bar.