# Domination versus edge domination

## TL;DR

This paper investigates the relationship between domination numbers and edge domination numbers in regular graphs, proposing a conjecture and providing bounds and verifications for specific graph classes.

## Contribution

It introduces a conjecture relating domination and edge domination numbers in regular graphs and proves bounds supporting this conjecture, including for cubic claw-free graphs.

## Key findings

- Proves bounds on the domination number relative to the edge domination number for regular graphs.
- Verifies the conjecture for cubic claw-free graphs.
- Provides specific bounds for graphs with degree 3.

## Abstract

We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph. We prove that $\gamma(G)\leq \left(1+\frac{2(\Delta-1)}{\Delta 2^{\Delta}}\right)\gamma_e(G)$ for general $\Delta\geq 1$, and $\gamma(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)\gamma_e(G)$ for $\Delta=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.