Approximating maximum-size properly colored forests

Bai, Yuhang, Bérczi, Kristóf, Csáji, Gergely and Schwarcz, Tamás (2026) Approximating maximum-size properly colored forests. European Journal of Combinatorics, 132. p. 104269. ISSN 0195-6698

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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 properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and -Traveling Salesman problems, two classical questions that have attracted considerable interest in combinatorial optimization and approximation theory. Previous work on properly colored spanning trees has mainly focused on determining the existence of such a tree and hence has not considered the question from an algorithmic perspective. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. Our proof technique relies on the sum of matching matroids defined by the color classes, a connection that might be of independent combinatorial interest. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.

Item Type:	Article
Additional Information:	© 2025 The Author's
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	27 Oct 2025 09:51
Last Modified:	27 Oct 2025 10:12
URI:	http://eprints.lse.ac.uk/id/eprint/129975

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