Hypergraph colouring

Csilla Bujtás; Zsolt Tuza; Vitaly Voloshin

doi:10.1017/CBO9781139519793.014

11 - Hypergraph colouring

Published online by Cambridge University Press: 05 May 2015

Csilla Bujtás ,

Zsolt Tuza and

Vitaly Voloshin

Edited by

Lowell W. Beineke and

Robin J. Wilson

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Csilla Bujtás: Affiliation:
University of Pannonia
Zsolt Tuza: Affiliation:
University of Pannonia
Vitaly Voloshin: Affiliation:
Troy University
Lowell W. Beineke: Affiliation:
Purdue University, Indiana
Robin J. Wilson: Affiliation:
The Open University, Milton Keynes

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Summary

We discuss the colouring theory of finite set systems. This is not merely an extension of results from collections of 2-element sets (graphs) to larger sets. The wider structure (hypergraphs) offers many interesting new kinds of problems, which either have no analogues in graph theory or become trivial when we restrict them to graphs.

Introduction

In this introductory section we give the most important definitions required to study hypergraph colouring, and briefly survey the half-century history of this topic. For more details on the material of Sections 1 and 2 we refer to Berge [8], Zykov [76] and Duchet [27].

Let V = {v1, v2, …, vn} be a finite set of elements called vertices, and let ℇ = {E1, E2, …, Em} be a family of subsets of V called edges or hyperedges. The pair ℌ = (V, ℇ) is called a hypergraph with vertex-set V = V(ℌ) and edge-set ℇ = ℇ(ℌ). The hypergraph ℌ = (V, ℇ) is sometimes called a set system. If each edge of a hypergraph contains precisely two vertices, then it is a graph. As in graph theory, the number |V| = n is called the order of the hypergraph. Edges with fewer than two elements are usually allowed, but will be disregarded here. Thus, throughout this chapter we assume that each edge E ∈ ℇ contains at least two vertices, unless stated explicitly otherwise. Edges that coincide are called multiple edges.

In a hypergraph, two vertices are said to be adjacent if there is an edge containing both of these vertices. The adjacent vertices are sometimes called neighbours of each other, and the set of neighbours of a given vertex v is called the (open) neighbourhood N(v) of v. If v ∈ E, then the vertex v and the edge E are incident with each other. For an edge E, the number |E| is called the size or cardinality of E.

Type: Chapter
Information: Topics in Chromatic Graph Theory , pp. 230 - 254

DOI: https://doi.org/10.1017/CBO9781139519793.014 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2015

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