Equidistant Paths in Graphs

Abstract

Let n and k be integers, n ≥ k ≥ 1. A graph G is said to admit property Pk if for any distinct pair x, y ∈ V (G), there exists k internally vertex disjoint paths between x and y of the same length. Consider the following family of graphs. G n k := {Gn : Gn admits property Pk}. There are two interesting directions in the study of G n k . Firstly, in the extremal direction, it is interesting to estimate the sparsity of graphs admitting property Pk. That is, estimation of ν(n, k) = min{|E(Gn)| : Gn ∈ Gn k }. The other direction is structural: what properties in the graph ensures admittance of property Pk. In this paper, we tackle the extremal question followed by some structural results on the same.