Lower Bounds for the Domination Numbers of Connected Graphs without Short Cycles

Abstract

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least $\frac{1}{2}(3+\sqrt{8(m-n)+9})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges in the graph. For graphs with minimum degree $2$ and girth at least $7$, the lower bound can be improved to $\max{\{\sqrt{n}, \sqrt{\frac{2m}{3}}\}}$, where $n$ and $m$ are the numbers of vertices and edges in the graph respectively. In cases where the graph is of minimum degree $2$ and its girth $g$ is at least $12$, the lower bound can be further improved to $\max{\{\sqrt{n}, \sqrt{\frac{\lfloor \frac{g}{3} \rfloor-1}{3}m}\}}$.