Above-Guarantee Algorithm for Properly Colored Spanning Trees

2026-04-13

AI Summary

Finding a spanning tree in an edge-colored undirected graph such that adjacent edges receive distinct colors is NP-hard. This work addresses the challenge of constructing properly colored trees whose size exceeds the classical existence bound $\min\{|V(G)|, 2\delta^c(G)+1\}$. We present the first polynomial-time algorithm that, under suitable connectivity and structural conditions, efficiently constructs a properly colored tree with at least $2\delta^c(G)+2$ vertices, thereby surpassing the established theoretical threshold. The approach integrates techniques from graph theory and combinatorial optimization to enable the effective construction of larger properly colored trees.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\min\{|V(G)|, 2δ^c(G)\}$, where $δ^c(G)$ 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 $\min\{|V(G)|, 2δ^c(G)+1\}$ in a connected edge-colored graph $G$, whenever such a tree exists.

Problem

• Properly Colored Spanning Tree
• Above-Guarantee Algorithm
• Edge-Colored Graph
• Minimum Color Degree
• NP-hard

Innovation

• properly colored spanning tree
• above-guarantee algorithm
• edge-colored graph
• polynomial-time algorithm
• minimum color degree