Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On an Erdős--Lov'asz problem: 3-critical 3-graphs of minimum degree 7

Ruiliang Li

TL;DR

This paper resolves Erdős–Lovász's question about the existence of a 3-critical 3-uniform hypergraph with minimum degree at least 7 by treating two non-equivalent notions of criticality. For the transversal interpretation, it proves a sharp bound: any $\tau$-critical 3-graph has at most $|E(H)|\le 10$, hence $\delta(H)\le 6$, with equality attained by the complete 3-graph $K^{(3)}_5$. In contrast, for the chromatic interpretation it constructs an explicit 3-uniform hypergraph on $9$ vertices with $\delta(H)=7$ that is critically 3-chromatic, along with exhaustive edge- and vertex-deletion certificates and a verification script. The results clarify the divergence between transversal and chromatic criticality in hypergraphs and provide reproducible, verifiable constructions for the chromatic case.

Abstract

Erdős and Lov'asz asked whether there exists a "3-critical" 3-uniform hypergraph in which every vertex has degree at least 7. The original formulation does not specify what 3-critical means, and two non-equivalent notions have appeared in the literature and in later discussions of the problem. In this paper we resolve the question under both interpretations. For the transversal interpretation (criticality with respect to the transversal number), we prove that a 3-uniform hypergraph $H$ with $τ(H)=3$ and $τ(H-e)=2$ for every edge $e$ has at most 10 edges; in particular, $δ(H)\le 6$, and this bound is sharp, witnessed by the complete 3-graph $K^{(3)}_5$. For the chromatic interpretation (criticality with respect to weak vertex-colourings), we give an explicit 3-uniform hypergraph on 9 vertices with $χ(H)=3$ and minimum degree $δ(H)=7$ such that deleting any single edge or any single vertex makes it 2-colourable. The criticality of the example is certified by explicit witness 2-colourings listed in the appendices, together with a short verification script.

On an Erdős--Lov'asz problem: 3-critical 3-graphs of minimum degree 7

TL;DR

This paper resolves Erdős–Lovász's question about the existence of a 3-critical 3-uniform hypergraph with minimum degree at least 7 by treating two non-equivalent notions of criticality. For the transversal interpretation, it proves a sharp bound: any -critical 3-graph has at most , hence , with equality attained by the complete 3-graph . In contrast, for the chromatic interpretation it constructs an explicit 3-uniform hypergraph on vertices with that is critically 3-chromatic, along with exhaustive edge- and vertex-deletion certificates and a verification script. The results clarify the divergence between transversal and chromatic criticality in hypergraphs and provide reproducible, verifiable constructions for the chromatic case.

Abstract

Erdős and Lov'asz asked whether there exists a "3-critical" 3-uniform hypergraph in which every vertex has degree at least 7. The original formulation does not specify what 3-critical means, and two non-equivalent notions have appeared in the literature and in later discussions of the problem. In this paper we resolve the question under both interpretations. For the transversal interpretation (criticality with respect to the transversal number), we prove that a 3-uniform hypergraph with and for every edge has at most 10 edges; in particular, , and this bound is sharp, witnessed by the complete 3-graph . For the chromatic interpretation (criticality with respect to weak vertex-colourings), we give an explicit 3-uniform hypergraph on 9 vertices with and minimum degree such that deleting any single edge or any single vertex makes it 2-colourable. The criticality of the example is certified by explicit witness 2-colourings listed in the appendices, together with a short verification script.
Paper Structure (23 sections, 13 theorems, 29 equations, 1 table)

This paper contains 23 sections, 13 theorems, 29 equations, 1 table.

Table of Contents

  1. Introduction
  2. Terminology and basic notions
  3. Hypergraphs, uniformity, and degrees
  4. Weak colourings, Property B, and chromatic criticality
  5. Transversals and tau-criticality of order 3
  6. The transversal interpretation: a sharp bound and the impossibility of delta>=7
  7. Bollobás's set-pairs inequality
  8. An edge bound for tau-critical 3-graphs
  9. A sharp minimum-degree consequence
  10. Sharpness: the extremal example
  11. The chromatic interpretation: a critically 3-chromatic 3-graph with delta=7
  12. The construction
  13. Degree computation
  14. H is not 2-colourable
  15. H is 3-colourable
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $H$ be a $3$-uniform hypergraph that is $\tau$-critical of order $3$, i.e. $\tau(H)=3$ and $\tau(H-e)=2$ for every $e\in E(H)$. Then $|E(H)|\le 10$ and consequently $\delta(H)\le 6$. Moreover, equality $\delta(H)=6$ is attained by the complete $3$-graph $K^{(3)}_5$.

Theorems & Definitions (28)

  • Theorem 1.1: Transversal interpretation
  • Theorem 1.2: Chromatic interpretation
  • Definition 2.1: Critically $3$-chromatic (weak)
  • Lemma 2.2: Edge-deletion certificates
  • proof
  • Definition 2.3: $\tau$-critical of order $3$
  • Lemma 3.1: Bollobás set-pairs inequality
  • proof
  • Theorem 3.2
  • proof
  • ...and 18 more