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Single conflict coloring and palette sparsification of uniform hypergraphs

Carl Johan Casselgren, Kalle Eriksson

TL;DR

This work extends single conflict coloring to $r$-uniform hypergraphs, clarifying its relationships with standard list-coloring variants and DP-coloring via foundational inequalities and constructions. It develops a probabilistic model using random local $k$-partitions, establishing a near-zero probability bound for small $k$, a degeneracy-based sufficient condition for whp colorability, and a sharp threshold result for complete graphs (with conjectured thresholds for complete $r$-uniform hypergraphs). Additionally, it proves a palette sparsification result for linear $r$-uniform hypergraphs, showing that sampling $k \ge A(\log n)^{1/r}$ colors per vertex from a palette of size $\sigma \ge C\Delta^{1/(r-1)}$ suffices whp, while highlighting that this does not extend to general hypergraphs. Overall, the paper broadens probabilistic methods and sparsification techniques from graphs to hypergraphs under single-conflict constraints, with implications for coloring in conflict-prone settings.

Abstract

We introduce and investigate single conflict coloring in the setting of r-uniform hypergraphs. We establish some basic properties of this hypergraph coloring model and study a probabilistic model of single conflict coloring where the conflicts for each edge are chosen randomly; in particular, we prove a sharp threshold-type result for complete graphs and establish a sufficient condition for single conflict colorability of r-uniform hypergraphs in this model. Furthermore, we obtain a related palette sparsification-type result for general list coloring of linear uniform hypergraphs (i.e. uniform hypergraphs where any two edges share at most one common vertex). Throughout the paper we pose several questions and conjectures

Single conflict coloring and palette sparsification of uniform hypergraphs

TL;DR

This work extends single conflict coloring to -uniform hypergraphs, clarifying its relationships with standard list-coloring variants and DP-coloring via foundational inequalities and constructions. It develops a probabilistic model using random local -partitions, establishing a near-zero probability bound for small , a degeneracy-based sufficient condition for whp colorability, and a sharp threshold result for complete graphs (with conjectured thresholds for complete -uniform hypergraphs). Additionally, it proves a palette sparsification result for linear -uniform hypergraphs, showing that sampling colors per vertex from a palette of size suffices whp, while highlighting that this does not extend to general hypergraphs. Overall, the paper broadens probabilistic methods and sparsification techniques from graphs to hypergraphs under single-conflict constraints, with implications for coloring in conflict-prone settings.

Abstract

We introduce and investigate single conflict coloring in the setting of r-uniform hypergraphs. We establish some basic properties of this hypergraph coloring model and study a probabilistic model of single conflict coloring where the conflicts for each edge are chosen randomly; in particular, we prove a sharp threshold-type result for complete graphs and establish a sufficient condition for single conflict colorability of r-uniform hypergraphs in this model. Furthermore, we obtain a related palette sparsification-type result for general list coloring of linear uniform hypergraphs (i.e. uniform hypergraphs where any two edges share at most one common vertex). Throughout the paper we pose several questions and conjectures
Paper Structure (4 sections, 18 theorems, 62 equations)

This paper contains 4 sections, 18 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Single conflict coloring of hypergraphs
  3. A probablistic model of single conflict coloring
  4. Palette sparsification of list coloring of hypergraphs

Key Result

Proposition 2.1

For any $r$-uniform hypergraph $H$ of average degree $d$, $\chi_{\nleftrightarrow}(H) \geq \left\lceil \left(\frac{d}{\log d}\right)^{1/r}\right\rceil$.

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof : Proof of Proposition \ref{['prop:eq']}
  • ...and 16 more