Broere, IzakHeidema, Johannes2012-09-192012-09-192013Broere, I & Heidema J 2012, 'Universal H-colorable graphs', Graphs and Combinatorics, vol. 29, no. 5, pp. 1193-1206.0911-0119 (print)1435-5914 (online)10.1007/s00373-012-1216-5http://hdl.handle.net/2263/19830Rado 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.en© Springer-Verlag 2012. The original publication is available at www.springerlink.com.Universal graphHom-property of graphsExtension property of graphsHomogeneous graphH-colourable graphk-colourable graph(k, l)-split graphRado graphPostprint Article