Random Graph-Homomorphisms and Logarithmic Degree

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function psi(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).