# Hypergraph colouring

Csilla Bujtás, Z. Tuza, Vitaly Voloshin

Research output: Chapter in Book/Report/Conference proceedingChapter

7 Citations (Scopus)

### Abstract

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.

Original language English Topics in Chromatic Graph Theory Cambridge University Press 230-254 25 9781139519793, 9781107033504 https://doi.org/10.1017/CBO9781139519793.014 Published - Jan 1 2015

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

Bujtás, C., Tuza, Z., & Voloshin, V. (2015). Hypergraph colouring. In Topics in Chromatic Graph Theory (pp. 230-254). Cambridge University Press. https://doi.org/10.1017/CBO9781139519793.014