$k$-regular bipartite domination number bound

Let $G$ be a $k$-regular bipartite graph with bipartition $V(G)=A\cup B$ and $|A|=|B|=n$. Let $\gamma(G)$ be the domination number of $G$, and let $\rho_G$ denote its normalised domination density. $k$-regular bipartite domination number bound. For every such graph,

$$\gamma(G)\leq 2\left\lceil \frac{n}{k}\right\rceil, \qquad \rho_G\leq\frac{1}{n}\left\lceil \frac{n}{k}\right\rceil.$$

The bound is proposed as a route toward proving Vizing's conjecture for balanced $k$-regular bipartite graphs. It is not proved in full generality and is used in the paper to reduce the remaining verification to finitely many unresolved cases.