Bounds and Hardness Results for Conflict-free Choosability

2024-09-19

AI Summary

This paper investigates the neighborhood conflict-free list coloring problem on hypergraphs: each vertex must select a color from its prescribed list such that, in every closed-neighborhood hyperedge, at least one vertex has a uniquely appearing color. We first extend the known $O(log^2 Delta)$ upper bound for closed-neighborhood conflict-free coloring to the list setting, proving that the closed-neighborhood conflict-free list chromatic number of any graph is at most $O(log^2 Delta)$. Second, via refined polynomial-time reductions, we establish that both the open- and closed-neighborhood variants of conflict-free list coloring are NP-hard. Technically, our approach integrates combinatorial probabilistic methods, hypergraph coloring theory, list-coloring analysis, and extremal graph-theoretic tools. This unified framework precisely characterizes the solvability threshold and computational complexity landscape of neighborhood conflict-free list coloring problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A'conflict-free coloring'of a hypergraph $mathcal{H}$ is an assignment of colors to the vertex set of $mathcal{H}$ such that every hyperedge in $mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the'conflict-free chromatic number'of $mathcal{H}$. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study the list variant of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. It was shown by Pach and Tardos [Combinatorics, Probability and Computing, 2009] that for any constant $epsilon>0$, the closed-neighborhood conflict-free chromatic number of a graph $G$ is at most $O(ln^{2 + epsilon}Delta)$, where $Delta$ represents the maximum degree of $G$. Later, Glebov, Szab'o, and Tardos [Combinatorics, Probability and Computing, 2014] showed that there exist graphs $G$ that require $Omega(ln^2Delta)$ colors for a closed neighborhood conflict-free coloring. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] bridged the gap between the upper and the lower bound. They showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ is at most $O(ln^2 Delta)$. In this paper, we extend the $O(ln^2 Delta)$ upper bound to the list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.

Problem

• Extends conflict-free coloring bounds to list variants
• Studies computational complexity of list conflict-free coloring
• Analyzes neighborhood hypergraphs with restricted color choices

Innovation

• Extends conflict-free coloring to list variant
• Proves O(ln²Δ) bound for list coloring
• Establishes computational complexity of list variants