A Short Note on the 1, 2-Good-Neighbor Diagnosability of Balanced Hypercubes

Abstract

Let tc(G) and tg(G) be the conditional diagnosability and g-good-neighbor diagnosability, respectively, of a graph G. The notion of the g-good-neighbor conditional diagnosability is less restrictive as compared with that of the conditional diagnosability in general. Particularly, the conditional faulty set notion requires that, any vertex, faulty or not, have at least one non-faulty neighbor; while the 1-good-neighbor faulty only requires that a non-faulty vertex have at least one non-faulty neighbor. Compared with conditional diagnosability, g-good-neighbor diagnosability is interesting since it characterizes a stronger tolerance capability. In this paper, we investigate the equal relation between t1(BHn) and tc(BHn) for the balanced hypercubes BHn. That is [Formula: see text] for [Formula: see text] under the PMC model and [Formula: see text] for [Formula: see text] under the MM model; Furthermore, the 2-good-neighbor diagnosability t2(BHn) = 4n − 1 for n ≥ 2 under the PMC model and the MM model is obtained.