Wide diameter and minimum length of disjoint Menger path systems

Abstract

Let k be a positive integer and let G be a graph with at least k +1 vertices. For two distinct vertices x , y of G, the k -wide distance d k (x , y ) between x and y is the minimum integer l such that there exist k internally disjoint (x , y )-paths whose lengths are at most l . We define d k (x , x ) = 0. The k -wide diameter d k (G) of G is the maximum value of the k -wide distances between two vertices of G. Let X , Y be k -subsets of V (G). We define m k (X , Y ) to be the minimum integer l such that there exist k vertex-disjoint (X , Y )-paths of length at most l , and we define m k (G) to be the maximum value of m k (X , Y ) over all k -subsets X , Y of V (G). We study relationships between d k (G) and m k (G). Among other results, we show that if k ≥ 2 and G is a k -connected graph, then

Node-disjoint paths in hierarchical hypercube networks

“…There are some interesting combinatorial and topological problems, e.g., Rabin number [4,14], wide diameter [2,13], hamiltonicity [7,8,18] and pancyclicity [5,26]. They are still open for the hierarchical hypercube network.…”

Wide-diameter of Product Graphs

“…For a discussion on the relation of wide diameters to disjoint path systems see [13]. For more results on the Rabin numbers we refer to [15,14].…”