The domination-number conjecture for total graphs of finite rings

Let $R$ be an arbitrary ring with Jacobson radical $J$, and suppose that

$$R/J=M_{n_1}(F_1) \times \ldots \times M_{n_k}(F_k),$$

where each $F_i$ is a finite field and $n_i\geq 2$ for all $i$. For a finite ring $A$, let $\tau(A)$ denote its total graph and let $\gamma(\tau(A))$ denote the domination number of that graph. Domination-number conjecture.

$$\gamma(\tau(R))=\min_i\{n_i(|F_i|-1)+1\}.$$

The preceding theorem establishes the corresponding upper bound without the restrictions $n_i\geq 2$. The examples of $M_2(\mathbb{Z}_2)$ and $M_3(\mathbb{Z}_2)$ attain this bound, while factors with $n_i=1$ can give a strictly smaller domination number; the equality in the stated generality is therefore proposed but not established here.