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Revealing Neural Circuit Topography in Multi-Color

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Published on: November 14, 2011

Hypergraph coloring complexes.

Felix Breuer¹, Aaron Dall, Martina Kubitzke

¹Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany.

Discrete Mathematics

|March 14, 2013

Summary

This summary is machine-generated.

This study generalizes graph coloring complexes to hypergraphs, finding many graph properties do not apply. New bounds for hypergraph chromatic polynomials are derived from the generalized coloring complex.

Keywords:

Chromatic polynomial Cohen–Macaulay Coloring complex Ehrhart theory Hypergraph Wedge lemma

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Area of Science:

Combinatorics
Algebraic Topology
Graph Theory

Background:

Coloring complexes are essential tools in graph theory.
Generalizing these complexes to hypergraphs presents unique challenges and opportunities.

Purpose of the Study:

To extend the concept of coloring complexes from graphs to hypergraphs.
To investigate the properties of these hypergraph coloring complexes.
To derive new bounds for hypergraph chromatic polynomials.

Main Methods:

Developed three interpretations of hypergraph coloring complexes: combinatorial and two geometric.
Analyzed the properties of these complexes, including Cohen-Macaulayness and partitionability.
Derived bounds for the - and -vectors of hypergraph coloring complexes.

Main Results:

Most graph coloring complex properties, such as Cohen-Macaulayness, do not generalize to hypergraphs.
Established new bounds for hypergraph chromatic polynomials using the generalized coloring complex.
Characterized hypergraphs for which the coloring complex is connected.
Provided an example showing hypergraph coloring complex decomposition is not always a wedge of spheres.

Conclusions:

The generalization of coloring complexes to hypergraphs reveals significant differences from their graph counterparts.
The study provides valuable new bounds for hypergraph chromatic polynomials.
Further research is needed to fully understand the topological properties of hypergraph coloring complexes.