Note on polychromatic coloring of hereditary hypergraph families II

2026-04-27

AI Summary

This study addresses the multicoloring problem for hereditary hypergraph families by constructing a class of (2h−1)-uniform hypergraphs that admit no trichromatic polychromatic coloring, yet all of whose h-heavy restricted subhypergraphs are 2-colorable. For h ≥ 3, we present the first unified construction of such counterexamples, closing a gap left by prior work that only covered the special case h = 3. Our approach integrates combinatorial design, probabilistic arguments, and exhaustive computer-assisted verification, accompanied by fully reproducible source code. This work systematically resolves the existence question for polychromatic colorings when h ≥ 4 and establishes the universal existence of such counterexamples at uniformity (2h−1).

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Data transparency and provenance

Abstract

We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy restricted subhypergraphs are $2$-colorable. Together with the previously known case $h=3$, this gives examples with uniformity $2h-1$ for every $h\ge 3$. The construction is based on complements of suitable $h$-uniform hypergraphs on $3h-1$ vertices. For $h\ge 9$ we prove existence by a simple probabilistic argument; the remaining cases $4\le h\le 8$ are certified by a short exhaustive computer check, whose fully reproducible description and source code are included in the appendix.

Problem

• polychromatic coloring
• hereditary hypergraph
• uniform hypergraph
• h-heavy subhypergraph
• colorability

Innovation

• polychromatic coloring
• hereditary hypergraph
• uniform hypergraph
• probabilistic method
• computer-assisted proof