Minimum neighborhood in a generalized cube

Authors:
Xiaofan Yang;Jianqiu Cao;Graham M. Megson;Jun Luo
Affiliations:
School of Computer and Information, Chongqing Jiaotong University, Chongqing, P.R. China and College of Computer Science, Chongqing University, Chongqing, P.R. China;School of Computer and Information, Chongqing Jiaotong University, Chongqing, P.R. China;Department of Computer Science, School of Systems Engineering, University of Reading, Reading, Berkshire, UK;College of Computer Science, Chongqing University, Chongqing, P.R. China
Venue:
Information Processing Letters
Year:
2006

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Abstract

Generalized cubes are a subclass of hypercube-like networks, which include some hypercube variants as special cases. Let θG(k) denote the minimum number of nodes adjacent to a set of k vertices of a graph G. In this paper, we prove θG(k) ≥ -1/2;k2 + (2n - 3/2)k - (n2 - 2) for each n-dimensional generalized cube and each integer k satisfying n + 2 ≤ k ≤ 2n. 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].