Countable α-extendable graphs

Abstract

Let us consider a countable graph G with vertex set V( G). Nash–Williams introduced the notion of an n-path, a 0- path is a finite path and for any n∈ N , an ( n+1)- path is a path P such that, for every finite subset F of V( G), P can be extended to an n-path containing F. This notion extends in a natural way to the concept of an α- path, where α is an ordinal. Polat proved that a countable graph which contains an ω 1-path has a hamiltonian path. The aim of this paper is to show that one cannot improve this theorem to an ordinal strictly less than ω 1: for any countable ordinal α, we exhibit a countable non-hamiltonian graph which contains an α-path. These graphs have maximal degree 4.