List coloring ordered graphs with forbidden induced subgraphs

2025-09-26

AI Summary

This paper investigates the computational complexity boundary of List $k$-Coloring—particularly List 4-Coloring—on ordered graphs that forbid a specific induced ordered subgraph. Adopting a systematic classification of all single-vertex-ordered forbidden patterns, the authors integrate structural graph-theoretic analysis, refined algorithm design, and polynomial-time reductions. They establish the first *almost complete complexity dichotomy* for List 4-Coloring on $H$-free ordered graphs: for nearly all ordered graphs $H$, they fully determine whether List 4-Coloring is polynomial-time solvable or NP-complete on the class of $H$-free ordered graphs, and uniquely identify a single minimal open case. This work delivers the most comprehensive dichotomy result to date for List 4-Coloring on ordered graphs, substantially advancing the theoretical understanding of coloring problems in the ordered graph setting.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

In the List $k$-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of ${1,ldots,k}$. We need to decide if $G$ admits a proper coloring, where every vertex receives a color from its list. The complexity of the problem in classes defined by forbidding induced subgraphs is a widely studied topic in algorithmic graph theory. Recently, Hajebi, Li, and Spirkl [SIAM J. Discr. Math. 38 (2024)] initiated the study of List $3$-Coloring in ordered graphs, i.e., graphs with fixed linear ordering of vertices. Forbidding ordered induced subgraphs allows us to investigate the boundary of tractability more closely. We continue this direction of research, focusing mostly on the case of List $4$-Coloring. We present several algorithmic and hardness results, which altogether provide an almost complete dichotomy for classes defined by forbidding one fixed ordered graph: our investigations leave one minimal open case.

Problem

• List 4-Coloring complexity for ordered graphs
• Dichotomy classification with forbidden induced subgraphs
• Resolving all but one minimal open case

Innovation

• List coloring with forbidden induced subgraphs
• Algorithmic and hardness dichotomy results
• Focus on ordered graphs with linear ordering