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

Let t(c)(G) and t(g)(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 t(1)(BHn) and t(c)(BHn) for the balanced hypercubes BHn. That is t(1)(BHn) = t(c)(BHn) = 4n for n >= 2 under the PMC model and t(1)(BHn) = t(c)(BHn) = 4n for n >= 2 under the MM model; Furthermore, the 2-good-neighbor diagnosability t(2)(BHn) = 4n - 1 for n >= 2 under the PMC model and the MM model is obtained.