The connection between the chromatic numbers of a hypergraph and its 1-intersection graph

Journal Article · Thu Sep 25 00:00:00 EDT 2025 · Discrete Mathematics

DOI:https://doi.org/10.1016/j.disc.2025.114810· OSTI ID:3003163

Blázsik, Zoltán L. ^[1]; Lemons, Nathan Wishard ^[2]

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1. Univ. of Szeged (Hungary). HUN-REN Alfréd Rényi Institute of Mathematics
2. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

A well known problem from an excellent book of Lovász states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh and of Gyárfás et al. we study the 1-intersection graph of a hypergraph. The 1-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for k ϵ {2, 4} that all hypergraphs whose 1-intersection graph is k-partite can be properly k-colored.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 3003163
• Report Number(s): LA-UR--24-26345; 10.1016/j.disc.2025.114810
• Journal Information: Discrete Mathematics, Journal Name: Discrete Mathematics Journal Issue: 2 Vol. 349; ISSN 0012-365X
• Publisher: Elsevier BV
• Country of Publication: United States
• Language: English

Related Subjects

1-intersection graph of hypergraphs
97 MATHEMATICS AND COMPUTING
Bipartite graph
Chromatic number
Greedy coloring
Recoloring process