Uniformly Connected Graphs — A Survey

Abstract

A graph [Formula: see text] of order [Formula: see text] is [Formula: see text]-uniformly connected for an integer [Formula: see text] with [Formula: see text] if for every pair [Formula: see text], [Formula: see text] of distinct vertices of [Formula: see text], there is a [Formula: see text] path of length [Formula: see text]. A number of results, conjectures, and problems are presented concerning [Formula: see text]-uniformly connected graphs for various integers [Formula: see text]. These include the special cases where [Formula: see text] and [Formula: see text]. Graphs are discussed that are [Formula: see text]-uniformly connected for a particular integer [Formula: see text] but are not [Formula: see text]-uniformly connected for every integer [Formula: see text]. Also, graphs are considered in which there is a unique [Formula: see text] path of length [Formula: see text] for a particular value of [Formula: see text]. Sets [Formula: see text] of positive integers are considered for which there exists a graph [Formula: see text] such that [Formula: see text] is [Formula: see text]-uniformly connected if and only if [Formula: see text].