# Strong domatic number of a graph

## TL;DR

This paper introduces the concept of the strong domatic number in graphs, establishing bounds and calculating it for specific graph classes, thereby advancing understanding of strong domination properties.

## Contribution

It initiates the study of the strong domatic number, providing bounds and exact values for certain classes of graphs, a novel contribution in graph domination theory.

## Key findings

- Established sharp bounds on the strong domatic number.
- Determined the strong domatic number for cubic graphs up to order 10.
- Introduced the concept of strong domatic number and initiated its systematic study.

## Abstract

A set $D$ of vertices of a simple graph $G=(V,E)$ is a strong dominating set, if for every vertex $x\in \overline{D}=V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. The strong domatic number of $G$ is the maximum number of strong dominating sets into which the vertex set of $G$ can be partitioned. We initiate the study of the strong domatic number, and we present different sharp bounds on $d_{st}(G)$. In addition, we determine this parameter for some classes of graphs, such as cubic graphs of order at most $10$.