On cubic rainbow domination regular graphs

A d -regular graph X is called d -rainbow domination regular or d -RDR, if its d -rainbow domination number γ r d ( 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.