The λ3-connectivity and κ3-connectivity of recursive circulants

Let S be a set of at least two vertices in a graph G. A subtree T of G is a S-Steiner tree if S subset of V(T). Two S-Steiner trees T-1 and T-2 are edge-disjoint (resp. internally disjoint) if E(T-1) boolean AND E(T-2) = empty set (resp. E(T-1) boolean AND E(T-2) = empty set and V(T-1) boolean AND V(T-2) = S). Let lambda(G)(S) (resp. kappa(G)(S)) be the maximum number of edge-disjoint (resp. internally disjoint) S-Steiner trees in G, and let lambda(k)(G) (kappa(k)(G)) be the minimum lambda(G)(S) (resp. kappa(G)(S)) for S ranges over all k-subsets of V(G). Clearly, lambda(2)(G) (resp. kappa(2)(G)) is the classical edge-connectivity lambda(G) (resp. connectivity kappa(G)). In this paper, we study the lambda(3)-connectivity and kappa(3)-connectivity of a recursive circulant G, determine lambda(3)(G) = delta(G) - 1 for each recursive circulant G, and kappa(3)(G) = delta(G) - 1 for each recursive circulant G except G congruent to G(2(m), 2). (C) 2018 Elsevier Inc. All rights reserved.