The g-good-neighbor diagnosability of triangle-free graphs

Abstract

Today’s supercomputers contain thousands of processing cores. As the number of processing units grows, the probability of failing some processing units increases. Therefore, finding faulty units has become a concern in these systems which depends on the diagnosability of the interconnection network that connects processing units. In this paper, we investigate the g-good-neighbor diagnosability of triangle-free graphs under the $\text {MM}^{*}$ and ${\text {PMC}\ }$ models. We show that if G is a connected triangle-free graph with minimum degree $\delta$, its diagnosability under the ${\text{MM}}^{*}$ model, i.e., $t^{\text {MM}^{*}}(G)$, is either $\delta -2$, $\delta -1$, or $\delta$. Also, the 1-good-neighbor diagnosability of G, i.e., $t_1^{\text {MM}^{*}}(G)$, is at least $\delta -1$ if it does not contain any subgraph isomorphic to $K_{\delta , \delta }$. Moreover, we show that if G does not contain a subgraph isomorphic to $K_{\delta -1, \delta -1},$ then it is 1-good-neighbor $\left( {\delta + 1} \right)$-diagnosable under $\mathrm{PMC}$ model when $\vert V(G)\vert > 2\delta +2$. Our results give lower bounds on the diagnosability and 1-good-neighbor diagnosability of triangle-free graphs which covers a broad class of interconnection networks.