Minimum neighborhood in a generalized cube

Abstract

Generalized cubes are a subclass of hypercube-like networks, which include some hypercube variants as special cases. Let θ G ( k ) denote the minimum number of nodes adjacent to a set of k vertices of a graph G. In this paper, we prove θ G ( k ) ⩾ − 1 2 k 2 + ( 2 n − 3 2 ) k − ( n 2 − 2 ) for each n-dimensional generalized cube and each integer k satisfying n + 2 ⩽ k ⩽ 2 n . Our result is an extension of a result presented by Fan and Lin [J. Fan, X. Lin, The t / k -diagnosability of the BC graphs, IEEE Trans. Comput. 54 (2) (2005) 176–184].