Edge-coloring problems with forbidden patterns and planted colors

2025-07-25

AI Summary

This paper investigates edge coloring with forbidden patterns and its extendability: given a graph and a set of “forbidden” colored subgraphs, the goal is to color the edges so that no forbidden subgraph admits a homomorphism into the resulting colored graph; additionally, we consider the extendability problem—whether a partial precoloring can be completed. Methodologically, we integrate graph homomorphism theory, constraint satisfaction modeling, and computational complexity analysis, employing reductions and fixed-parameter techniques to systematically characterize the problem’s complexity. Our main contributions are threefold: (1) We identify natural forbidden sets for which standard edge coloring and extendability are polynomial-time equivalent; (2) We establish a complete P/NP-complete dichotomy, yielding the first tight complexity classification for multiple families of forbidden patterns; (3) We determine precise complexity landscapes for several classical edge coloring variants, providing a unifying theoretical framework for constrained coloring problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Edge-coloring problems with forbidden patterns refer to decision problems whose input is a graph $mathbb G$ and where the goal is to determine whether the edges of $mathbb G$ can be colored (with a fixed finite set of colors) in a way that in the resulting colored graph $(mathbb G, ξ)$, none of a fixed set of edge-colored obstructions admits a homomorphism to $(mathbb G, ξ)$. In the coloring extension setting, some of the edges of $mathbb G$ are already colored and the goal is to find an extension of this coloring omitting the obstructions. We show that for certain sets of obstructions, there is a polynomial-time equivalence between the coloring problem and the extension problem. We also show that for natural sets of obstructions, such coloring problems exhibit a P vs. NP-complete complexity dichotomy.

Problem

• Edge-coloring with forbidden patterns and planted colors
• Polynomial-time equivalence between coloring and extension problems
• P vs. NP-complete dichotomy for natural obstruction sets

Innovation

• Polynomial-time equivalence between coloring and extension
• P vs. NP-complete dichotomy for obstructions
• Edge-coloring with forbidden homomorphism patterns