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.
We motivate, introduce, and study radicals on classes of graphs. This
concept, and the theory which is developed, imitates the original notion
of a Hoehnke radical in universal algebra using congruences. It is shown
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. ...
We extend results about asymmetric colorings of finite Cartesian products of graphs to strong and direct products of graphs and digraphs. On the way we shorten proofs for the existence of prime factorizations of finite ...