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 \(2\delta ^{col}(G) \ne |G|\).

边着色图G是具有边着色的图。我们说摹正确彩色如果任何两个相邻的边摹具有不同的颜色，并摹是彩虹如果任何两条边摹具有不同的颜色。对于一个顶点（以V v \（G）\）\，色彩度\（D_G ^ {COL}（V）\）的v是出现在边缘入射具有不同颜色的数目v。最小颜色程度\（\三角洲^ {COL}（G）\）的ģ是最小\（D_G ^ {COL}（V）\）对所有顶点\（V \在V（G）\）. 在本文中，我们研究了最大妥善彩色树的顺序之间的关系摹和最小色度\（\三角洲^ {}山口（G）\）的摹。我们得到，对于边着色连通图G，最大适当着色树的阶数至少为\(\min \{|G|, 2\delta ^{col}(G)\}\)，这概括了Cheng 等人的结果。[边彩色图中正确着色的生成树，离散数学，343 (1), 2020]。此外，下界\（2 \增量^ {COL}（G）\）在我们的结果是锋利的，我们表征所有极图ģ用的顺序的最大正常着色树\（2 \增量^ {COL}（G ) \ne |G|\)。