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Coloring [Formula: see text]-Embeddable [Formula: see text]-Uniform Hypergraphs.

Carl Georg Heise1, Konstantinos Panagiotou2, Oleg Pikhurko3

  • 1Institut für Mathematik, Technische Universität Hamburg-Harburg, Hamburg, Germany.

Discrete & Computational Geometry
|November 25, 2014
PubMed
Summary
This summary is machine-generated.

This study explores hypergraph coloring, extending the Four Color Theorem. Researchers established bounds for the chromatic number of specific hypergraphs, providing new insights into graph theory.

Keywords:
Four Color TheoremChromatic numberColoringEmbeddingsHypergraphs

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Area of Science:

  • Graph Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • The Four Color Theorem addresses coloring planar graphs.
  • Hypergraphs generalize graphs by allowing edges to connect multiple vertices.
  • Understanding hypergraph coloring is crucial for various applications.

Purpose of the Study:

  • To extend the principles of the Four Color Theorem to hypergraphs.
  • To investigate the maximum weak chromatic number of hypergraphs embeddable in a specific surface.
  • To establish lower and upper bounds for this chromatic number.

Main Methods:

  • Definition of [Formula: see text]-uniform hypergraphs embeddable in [Formula: see text].
  • Analysis of chromatic numbers for these hypergraphs.
  • Derivation of specific bounds based on vertex count.

Main Results:

  • Demonstrated existence of hypergraphs in [Formula: see text] with a chromatic number of [Formula: see text] on [Formula: see text] vertices.
  • Established an upper bound of [Formula: see text] for the chromatic number of [Formula: see text]-vertex hypergraphs in [Formula: see text].

Conclusions:

  • The chromatic number of hypergraphs exhibits complex behavior related to their structure and embeddability.
  • This research provides a foundation for further studies in hypergraph coloring and its bounds.