Above-Guarantee Algorithm for Properly Colored Trees

Abstract

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. While it is known that every connected edge-colored graph $G$ contains a properly colored tree of order at least $\min \{|V(G)|,2\hat{I}{´}^c(G)\}$, where $\hat{I}{´}^c(G)$ 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 $\min \{|V(G)|,2\hat{I}{´}^c(G)+1\}$ in a connected edge-colored graph $G$, whenever such a tree exists.