Above-Guarantee Algorithm for Properly Colored Spanning Trees

TL;DR

This paper presents a polynomial-time algorithm that constructs large properly colored spanning trees in edge-colored graphs, surpassing previous guarantees based on minimum color degree conditions.

Contribution

The authors develop an above-guarantee polynomial-time algorithm for finding large properly colored spanning trees, improving upon existing bounds.

Findings

The algorithm guarantees a properly colored spanning tree of size at least min{|V(G)|, 2δ^c(G)+1} when such a tree exists.

The approach extends previous bounds by providing an above-guarantee construction.

The method applies to connected edge-colored graphs and is polynomial-time.

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 $\min\{|V(G)|, 2\delta^c(G)\}$, 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.