Universal H-colourable 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 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.http://www.springerlink.com/content/0911-011

Distinguishing homomorphisms of infinite graphs

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core

Universality for and in induced-hereditary graph properties

The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 20 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(20 ) properties in the lattice K< of induced-hereditary properties of which only at most 20 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.http://www.discuss.wmie.uz.zgora.pl/gt/am201

Universal H-Colourable 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 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.http://www.springerlink.com/content/0911-011