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Note on Polychromatic Coloring of Hereditary Hypergraph Families.

Dömötör Pálvölgyi¹

¹ELTE Eötvös Loránd University and Alfréd Rényi Institute of Mathematics, Budapest, Hungary.

Graphs and Combinatorics

|November 25, 2024

Summary

Researchers constructed a 5-uniform hypergraph lacking a polychromatic 3-coloring, disproving a conjecture. This finding advances the understanding of polychromatic colorings in hereditary hypergraph families.

Keywords:

Graph colorings Hypergraphs Panchromatic coloring Polychromatic coloring

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Area of Science:

Combinatorics
Graph Theory
Hypergraph Theory

Background:

Polychromatic coloring is a concept in hypergraph theory.
Hereditary hypergraph families are collections of hypergraphs closed under subgraphs.
Previous work by Berge laid foundational understanding in this area.

Purpose of the Study:

To disprove a conjecture by Keszegh and the author regarding polychromatic colorings.
To explore the properties of polychromatic colorings in specific hypergraph families.
To investigate the limitations of current methods for constructing such hypergraphs.

Main Methods:

Construction of a specific 5-uniform hypergraph.
Analysis of its polychromatic 3-colorability.
Examination of the 2-colorability of its restricted subhypergraphs (edges of size at least 3).

Main Results:

A 5-uniform hypergraph was exhibited that does not possess a polychromatic 3-coloring.
All restricted subhypergraphs with edges of size at least 3 were shown to be 2-colorable.
The presented method was found to be insufficient for generating hypergraphs of arbitrary high uniformity.

Conclusions:

The conjecture by Keszegh and the author is disproven.
This work represents a significant step in understanding polychromatic colorings of hereditary hypergraph families.
Connections to panchromatic colorings were also noted.