Optimal linear-Vizing relationships for (total) domination in graphs

A total dominating set in a graph $G$ is a set of vertices of $G$ such that every vertex is adjacent to a vertex of the set. The total domination number ${\gamma }_t\left(G\right)$ is the minimum cardinality of a total dominating set in $G$. In this paper, we study the following open problem posed by Yeo. For each $\Delta \ge 3$, find the smallest value, $r_{\Delta }$, such that every connected graph $G$ of order at least 3, of order $n$, size $m$, total domination number ${\gamma }_t$, and bounded maximum degree $\Delta$, satisfies $m\le \frac{1}{2}\left(\Delta +r_{\Delta }\right)\left(n-{\gamma }_t\right)$. Henning showed that $r_{\Delta }\le \Delta$ for all $\Delta \ge 3$. Yeo significantly improved this result and showed that $0.1\ln \left(\Delta \right)<r_{\Delta }\le 2\sqrt{\Delta }$ for all $\Delta \ge 3$, and posed as an open problem to determine “whether $r_{\Delta }$ grows proportionally with $\ln \left(\Delta \right)$ or $\sqrt{\Delta }$ or some completely different function.” In this paper, we determine the growth of $r_{\Delta }$, and show that $r_{\Delta }$ is asymptotically $\ln \left(\Delta \right)$ and likewise determine the asymptotics of the analogous constant for standard domination.