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

TL;DR

This paper establishes new lower bounds for the domination numbers of connected graphs with large girth, depending on parameters like number of vertices, edges, and minimum degree, advancing understanding of graph domination properties.

Contribution

It introduces novel lower bounds for domination numbers in connected graphs with girth at least 7, including improvements for graphs with minimum degree 2 and girth at least 12.

Findings

Lower bound for domination number: 1 or at least (1/2)(3+√(8(m-n)+9))

Improved bound for minimum degree 2: max{√n, √(2m/3)}

Further improved bound for girth ≥ 12: max{√n, √(((⌊g/3⌋-1)/3)m)}

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 $\max{\{\sqrt{n}, \sqrt{\frac{\lfloor \frac{g}{3} \rfloor-1}{3}m}\}}$.