Maximum properly colored trees in edge-colored graphs

Abstract

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 $v \in V(G)$, 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 $v \in V(G)$. 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 $\min \{|G|, 2\delta ^{col}(G)\}$, 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 $2\delta ^{col}(G) \ne |G|$.