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arXiv · 2604.11326

Above-Guarantee Algorithm for Properly Colored Trees

Yuhang Bai·Kristóf Bérczi
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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.

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Yuhang Bai, Kristóf Bérczi. 2026-04-13. Above-Guarantee Algorithm for Properly Colored Trees. https://arxiv.org/abs/2604.11326

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