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

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