Graph properties and hypergraph colourings

Abstract

Given a graph property P, graph G and integer k⩾0, a P k-colouring of G is a function π: V( G)→{1,..., k} such that the subgraph induced by each colour class has property P. When P is closed under induced subgraphs, we can construct a hypergraph H P G such that G is P k-colourable iff H P G is k-colourable. This correlation enables us to derive interesting new results in hypergraph chromatic theory from a ‘graphic’ approach. In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic numbers.