The k-Tuple Domatic Number of a Graph

TL;DR

This paper introduces the concept of the $k$-tuple domatic number in graphs, generalizing the well-known domatic number, and explores its fundamental properties and bounds.

Contribution

It defines the $k$-tuple domatic number, extends existing graph domination concepts, and derives initial properties and bounds for this new parameter.

Findings

Established basic properties of the $k$-tuple domatic number

Derived bounds for the $k$-tuple domatic number

Connected the $k$-tuple domatic number to existing domination parameters

Abstract

For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$. The minimum cardinality of a $k$-tuple dominating set of $G$ is the $k$-tuple domination number of $G$. When $k=1$, a $k$-tuple domination number is the well-studied domination number. We define the $k$-tuple domatic number of $G$ as the largest number of sets in a partition of $V$ into $k$-tuple dominating sets. Recall that when $k=1$, a $k$-tuple domatic number is the well-studied domatic number. In this work, we derive basic properties and bounds for the $k$-tuple domatic number.