# On k-rainbow domination in regular graphs

## TL;DR

This paper investigates the $k$-rainbow domination number in regular graphs, establishing bounds, necessary conditions for extremal cases, and exact values for specific Cayley graphs, advancing understanding of domination parameters in graph theory.

## Contribution

It provides a lower bound for the $k$-rainbow domination number in $d$-regular graphs and characterizes when this bound is tight, including exact values for certain Cayley graphs.

## Key findings

- Lower bound for $ ext{γ}_{rk}(G)$ in $d$-regular graphs
- Necessary conditions for graphs to attain the bound
- Exact $k$-rainbow domination numbers for cubic Cayley graphs

## Abstract

The $k$-rainbow domination problem is studied for regular graphs. We prove that the $k$-rainbow domination number $\gamma_{rk}(G)$ of a $d$-regular graph for $d\leq k\leq 2d$ is bounded below by $\displaystyle{\left\lceil kn/2d\right\rceil}$, where $n$ is the order of a graph. We determine necessary conditions for regular graphs to attain this bound and find several examples. As an application, we determine exact $k$-rainbow domination numbers for all cubic Cayley graphs over abelian groups.