The generalized 4-connectivity of exchanged hypercubes

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}. The parameter is a generalization of traditional connectivity. So far, almost all known results of kappa(r)(G) are about regular graphs and r = 3. In this paper, we focus On kappa(r)(EH(s, t)) of the exchanged hypercube for r = 4, where the exchanged hypercube EH(s, t) is not regular if s not equal t. We show that kappa(4)(EH(s, t)) = min{s, t} for min{s, t} >= 3. As a corollary, we obtain that kappa(3)(EH(s, t)) = min {s, t} for min{s, t} >= 3. (C) 2018 Elsevier Inc. All rights reserved.