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