In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a spanning tree in which any two adjacent edges have distinct colors. Since finding such a tree is NP-hard in general, previous work often relied on minimum color degree conditions to guarantee the existence of properly colored spanning trees. It is known that every connected edge-colored graph $G$ contains a properly colored tree of order at least $\min\{|V(G)|, 2\delta^c(G)\}$, where $\delta^c(G)$ denotes the minimum number of colors incident to a vertex. We study the algorithmic above-guarantee problem for properly colored trees. We provide a polynomial-time algorithm that constructs a properly colored tree of order at least $\min\{|V(G)|, 2\delta^c(G)+1\}$ in a connected edge-colored graph $G$, whenever such a tree exists.
Blogger's Review: This algorithm presents a novel approach to the properly colored tree problem, surpassing traditional constraints and demonstrating the potential and complexity of edge-colored graphs. The polynomial-time construction algorithm significantly enhances the feasibility of practical applications, warranting further exploration and practice.