Infinite Hamiltonian paths in Cayley digraphs

Abstract

Let Cay( S: H) be the Cayley digraph of the generators S in the group H. A one-way infinite Hamiltonian path in the digraph G is a listing of all the vertices [ ν i :1⩽ i<∞], such that there is an arc from ν i to ν 1+1. A two-way infinite Hamiltonian path is similarly defined, with i ranging from −∞ to ∞. In this paper, we give conditions on S and H for the existence of one- and two-way infinite Hamiltonian paths in Cay( X: H). Two of our results can be summarized as follows. First, if S is countably infinite and H is abelian, then Cay( S: H) has one- and two-way Hamiltonian paths if and only if it is strongly connected (except for one infinite family). We also give necessary and sufficient conditions on S for Cay( S: H) to be strongly connected for a large class of Cayley digraphs. Second, we show that any Cayley digraph of a countable locally finite group has both one- and two-way infinite Hamiltonian paths. As a lemma, we give a relation between the strong connectivity and the outer valence of finite vertex-transitive digraphs.