Distinguishing number and adjacency properties

Abstract

One of the most widely studied infinite graphs is the Rado or infinite random graph, written R. A graph satisfies the existentially closed or e.c. adjacency property if for all finite disjoint sets of vertices A and B (one of which may be empty), there is a vertex z / ∈ A ∪ B joined to all of A and to no vertex of B. By a back-and-forth argument, R is the unique isomorphism type of countably infinite graphs that is e.c. Further, R is homogeneous: every isomorphism between finite induced subgraphs extends to an automorphism. For a survey of these and other results on R, see [3]. The distinguishing number of a graph G, written D(G), is the least positive integer n such that there exists an n-colouring of V (G) (not necessarily proper) so that no non-trivial automorphism preserves the colours. Rigid graphs (which possess no non-trivial automorphisms) have distinguishing number 1, and D(G) may be viewed as the minimum number of colours needed to make G rigid. The parameter D(G) was introduced by Albertson and Collins [1]. The distinguishing number of graphs generalizes in a straightforward fashion to relational structures. A relation on a setX is a set of n-tuples from