Bounds and Extrema for Classes of Graphs and Finite Structures

Abstract

We consider the homomorphism (or colouring) order C induced by all finite structures (of a given type; for example graphs) and the existence of a homomorphism between them. This ordering may be seen as a lattice which is however far from being complete. In this paper we study (upper) bounds, suprema and maximal elements in C of some frequently studied classes of structures (such as classes of structures with bounded degree of its vertices, degenerated and classes determined by a finite set of forbidden substructures). We relate these extrema to cuts and duality theorems for C. Some of these results hold for general finite relational structures. In view of combinatorial problems related to coloring problems this should be regarded as a surprise. We support this view also by showing both analogies and striking differences between undirected and oriented graphs (i.e. for the easiest types) This is based on our recent work with C. Tardif.