On cubic rainbow domination regular graphs

TL;DR

This paper studies a special class of regular graphs called $d$-rainbow domination regular graphs, introduces methods to construct new examples, and classifies small vertex-transitive instances based on their structural properties.

Contribution

It provides new combinatorial constructions and criteria for identifying $d$-RDR graphs, along with a partial classification of small vertex-transitive $3$-RDR graphs.

Findings

Two construction methods for $d$-RDR graphs are described.

Criteria for vertex-transitive $d$-RDR graphs are proven.

A classification of small vertex-transitive $3$-RDR graphs is provided.

Abstract

A $d$-regular graph $X$ is called $d$-rainbow domination regular or $d$-RDR, if its $d$-rainbow domination number $\gamma_{rd}(X)$ attains the lower bound $n/2$ for $d$-regular graphs, where $n$ is the number of vertices. In the paper, two combinatorial constructions to construct new $d$-RDR graphs from existing ones are described and two general criteria for a vertex-transitive $d$-regular graph to be $d$-RDR are proven. A list of vertex-transitive 3-RDR graphs of small orders is produced and their partial classification into families of generalized Petersen graphs, honeycomb-toroidal graphs and a specific family of Cayley graphs is given by investigating the girth and local cycle structure of these graphs.