COLOURING—VARIATIONS ON A THEME

Abstract

This chapter sets out certain particular classes of homomorphism problems that have been investigated as variants of graph colourings. The homomorphism perspective unifies these concepts and offers new questions. The chapter includes a discussion of the circular chromatic number, the fractional chromatic number, the $T$-span, and the oriented chromatic number. Highlights include a number of equivalent definitions of the circular chromatic number in terms of $H$-colourability; in terms of a geometric representation, in terms of orientations implying, for instance, Minty’s result on chromatic numbers; and in terms of schedule concurrency. For fractional chromatic numbers, equivalent formulations are given in terms of Kneser graphs, integer linear programs, and zero-sum games, and they are related in several ways to the circular chromatic numbers. Homomorphisms amongst Kneser graphs are investigated, and a proof of Kneser’s conjecture is given. It is shown that the span for any set $T$ of the cliques $K_n$ has a limit, which is closely related to the fractional chromatic number of an associated graph. Bounds on the oriented chromatic numbers of planar and outerplanar graphs are given, and oriented chromatic numbers are related to acyclic chromatic numbers.