The generalized 4-connectivity of hierarchical cubic networks

Let S subset of V(G) and kappa(G)(S) denote the maximum number k of edge-disjoint trees T-1, T-2, ..., T-k in G such that V(T-i)boolean AND V(T-j) = S for any i, j is an element of {1, 2, ..., k} and i not equal j. For an integer r with 2 <= r <= n, the generalized r-connectivity of a graph G is defined as kappa(r)(G) = min{kappa(G) (S)vertical bar S subset of V(G) and vertical bar S vertical bar= r}. In fact, kappa(2)(G) is exactly the traditional connectivity of G. In this paper, we focus on kappa(4)(HCNn) of the hierarchical cubic network HCNn and obtain that kappa(4)(HCNn) = n for n >= 3. As a corollary, we obtain that kappa(3)(HCNn) = n for n >= 3. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This paper focuses on the generalized 4-connectivity of the hierarchical cubic network HCNn and shows that kappa(4)(HCNn) = n for n >= 3. As a corollary, it is also obtained that kappa(3)(HCNn) = n for n >= 3.