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Coloring one-headed directed hypergraphs

Balázs István Szabó

TL;DR

The paper proves the Keszegh–Pálvölgyi conjecture for the case of one-headed directed hypergraphs with edge size at least 3 by presenting a constructive 2-coloring algorithm that prevents any monochromatic edge. It then provides a general sufficient condition for 3-colorability in directed hypergraphs and derives related bounds via partitioning edges into head and tail categories. In the study of 2→1 hypergraphs, it shows that certain two-edge configurations do not bound chromatic numbers, giving constructions with arbitrarily large χ while avoiding them, and it establishes sufficiency results for 4-colorability via avoiding $I_0$ and for 2-colorability via avoiding both $I_0$ and $R_4$, complemented by explicit small examples. The work also highlights open questions about colorability when edges have multiple heads and whether avoidance of $I_0$ implies 3-colorability, indicating directions for future research in hypergraph coloring and structure theory.

Abstract

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and Pálvölgyi posed the following conjecture. Let $H$ be a directed hypergraph such that in every hyperedge the number of head-vertices is less than the number of tail-vertices and assume that for every pair of hyperedges $e_{1},e_{2}\in E(H)$ with $|e_{1}\cap e_{2}|=1$, the common vertex is a head-vertex in at least one of the hyperedges. Then $H$ admits a proper 2-coloring. Keszegh showed that the conjecture is also true in the special case of 3-uniform hypergraphs. A directed hypergraph is called one-headed if every hyperedge has exactly one head-vertex. The main result of this paper is that the conjecture is true for one-headed directed hypergraphs with all hyperedges having size at least three. Directed 3-uniform hypergraphs such that in every hyperedge the number of head-vertices is one and the number of tail-vertices is two are called $2\rightarrow 1$ hypergraphs. In this paper we consider sufficient conditions for $2\rightarrow 1$ hypergraphs to be proper $k$-colorable for some small $k$.

Coloring one-headed directed hypergraphs

TL;DR

The paper proves the Keszegh–Pálvölgyi conjecture for the case of one-headed directed hypergraphs with edge size at least 3 by presenting a constructive 2-coloring algorithm that prevents any monochromatic edge. It then provides a general sufficient condition for 3-colorability in directed hypergraphs and derives related bounds via partitioning edges into head and tail categories. In the study of 2→1 hypergraphs, it shows that certain two-edge configurations do not bound chromatic numbers, giving constructions with arbitrarily large χ while avoiding them, and it establishes sufficiency results for 4-colorability via avoiding and for 2-colorability via avoiding both and , complemented by explicit small examples. The work also highlights open questions about colorability when edges have multiple heads and whether avoidance of implies 3-colorability, indicating directions for future research in hypergraph coloring and structure theory.

Abstract

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and Pálvölgyi posed the following conjecture. Let be a directed hypergraph such that in every hyperedge the number of head-vertices is less than the number of tail-vertices and assume that for every pair of hyperedges with , the common vertex is a head-vertex in at least one of the hyperedges. Then admits a proper 2-coloring. Keszegh showed that the conjecture is also true in the special case of 3-uniform hypergraphs. A directed hypergraph is called one-headed if every hyperedge has exactly one head-vertex. The main result of this paper is that the conjecture is true for one-headed directed hypergraphs with all hyperedges having size at least three. Directed 3-uniform hypergraphs such that in every hyperedge the number of head-vertices is one and the number of tail-vertices is two are called hypergraphs. In this paper we consider sufficient conditions for hypergraphs to be proper -colorable for some small .

Paper Structure

This paper contains 8 sections, 10 theorems, 3 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['one_head']}
  3. A sufficient condition for 3-colorability of directed hypergraphs
  4. Coloring 2 $\rightarrow$ 1 hypergraphs
  5. Avoiding hypergraphs $H_{2}$, $I_{1}$, $R_{3}$ and $E$
  6. Avoiding the hypergraphs $I_{0}$ and $R_{4}$
  7. Open questions
  8. Acknowledgment

Key Result

Theorem 1.1

5 Let $H$ be a hypergraph in which every pair of hyperedges has an empty intersection or intersects in at least two vertices. Then $H$ admits a proper 2-coloring.

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3: The 3-chromatic hypergraph $I$, which avoids $I_{0}$
  • Figure 4: The 3-chromatic hypergraph $R$, which avoids $R_{4}$

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Claim 1.6
  • Claim 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 18 more