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.
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 m'th position of its binary expansion. ...
A graph property is a set of (countable) graphs. A homomorphism from a
graph G to a graph H is an edge-preserving map from the vertex set of G into
the vertex set of H; if such a map exists, we write G → H. Given any ...