Constricting the Computational Complexity Gap of the $4$-Coloring Problem in $(P_t,C_3)$-free Graphs

2025-09-02

AI Summary

This paper investigates the computational complexity of 4-coloring $(P_t,C_3)$-free graphs, bridging the gap between polynomial-time solvability (known for $t=6$) and NP-completeness (established for $t=22$). Using intricate graph-theoretic reductions, structural analysis of induced subgraphs, and computer-assisted verification, we prove that 4-coloring is NP-complete for $(P_{19},C_3)$-free graphs—the first such hardness result for $t geq 19$. We further extend this to $t = 20$ and $t = 21$. Consequently, the interval of unknown complexity is reduced from $t in [7,21]$ to $t in [7,18]$, significantly advancing the understanding of the coloring threshold in triangle-free graphs excluding long induced paths. This work provides the first rigorous NP-completeness result for $t geq 19$, sharpening the boundary between tractable and intractable instances within this fundamental graph class.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Web data visualization

Abstract

The $k$-Coloring problem on hereditary graph classes has been a deeply researched problem over the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs. We say that a graph is $(H_1,H_2,ldots)$-free if it does not contain any of $H_1,H_2,ldots$ as induced subgraphs. The complexity landscape of the problem remains unclear even when restricting to the case $k=4$ and classes defined by a few forbidden induced subgraphs. While the case of only one forbidden induced subgraph has been completely resolved lately, the complexity when considering two forbidden induced subgraphs still has a couple of unknown cases. In particular, $4$-Coloring on $(P_6,C_3)$-free graphs is polynomial while it is NP-hard on $(P_{22},C_3)$-free graphs. We provide a reduction showing NP-completeness of $4$-Coloring on $(P_t,C_3)$-free graphs for $19leq tleq 21$, thus constricting the gap of cases whose complexity remains unknown. Our proof includes a computer search ensuring that the graph family obtained through the reduction is indeed $P_{19}$-free.

Problem

• Resolving 4-coloring complexity in (P_t, C_3)-free graphs
• Determining computational tractability boundary for graph coloring
• Narrowing NP-completeness gap for specific hereditary graph classes

Innovation

• Reduction proving NP-completeness for 4-coloring
• Focuses on (P_t,C_3)-free graphs where t=19-21
• Uses computer search to verify P_19-free property