Universal H-colourable graphs

19 Sep 2012

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: for given m and n with m < n, m is adjacent to n if n has a 1 in the mth position of its binary expansion. It is well known that R is a universal graph in the set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph of R) and that it is a homogeneous graph (since every isomorphism between two finite induced subgraphs of R extends to an automorphism of R). In this paper we construct a graphU(H) which is H-universal in →Hc, the induced-hereditary hom-property of H-colourable graphs consisting of all (countable) graphs which have a homomorphism into a given (countable) graph H. If H is the (finite) complete graph Kk , then→Hc is the property of k-colourable graphs. The universal graph U(H) is characterised by showing that it is, up to isomorphism, the unique denumerable, H-universal graph in →Hc which is H-homogeneous in →Hc. The graphs H for which U(H) ∼= R are also characterised.With small changes to the definitions, our results translate effortlessly to hold for digraphs too. Another slight adaptation of our work yields related results for (k, l)-split graphs.