Gallai-Milgram properties for infinite graphs

Abstract

We discuss extensions of the Gallai-Milgram theorem to infinite graphs. We define a path to be a directed graph whose transitive closure is a linear ordering. We show that an undirected graph with no infinite independent set is covered by finitely many pairwise disjoint paths; moreover for a given integer k this graph is covered by ⩽k (resp. ⩽k pairwise disjoint) paths if each finite set of vertices is contained in the union of ⩽k (resp. ⩽k pairwise disjoint) paths. Hence an undirected graph with no independent set of size k + 1 is covered by ⩽k pairwise disjoint paths. We prove also that if to each edge (x, y) of a countable path is associated an element θ(x, y) of a finite group, then some edges can be deleted so that the new graph is still a path and the product θ(x0, x1)•θ(x1, x2)•⋯•θ(xn−1, xn) of the elem ents of the group along any finite path (x0, x1,…,xn) of the new graph depends only upon the extremities x0, x>n of the path.