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On cubic rainbow domination regular graphs

Bostjan Kuzman

Abstract

A $d$-regular graph $X$ is called $d$-rainbow domination regular or $d$-RDR, if its $d$-rainbow domination number $γ_{rd}(X)$ 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.

On cubic rainbow domination regular graphs

Abstract

A -regular graph is called -rainbow domination regular or -RDR, if its -rainbow domination number attains the lower bound for -regular graphs, where is the number of vertices. In the paper, two combinatorial constructions to construct new -RDR graphs from existing ones are described and two general criteria for a vertex-transitive -regular graph to be -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.

Paper Structure

This paper contains 9 sections, 15 theorems, 12 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Some general observations on $d$-RDR graphs and their colorings
  3. Vertex-transitive $d$-RDR graphs
  4. Vertex-transitive 3-RDR graphs
  5. Generalized Petersen Graphs
  6. Honeycomb toroidal graphs
  7. A new family of 3-RDR Cayley graphs
  8. Concluding remarks
  9. Acknowledgments

Key Result

Theorem 2.1

Let $X$ be a $d$-regular graph of order $n$ and $k\leq 2d$. Then:

Figures (4)

  • Figure 1: The three non-isomorphic 3-RDR graphs of order 12. The graph on the right is vertex-transitive and isomorphic to the prism $\mathop{\mathrm{Pr}}\nolimits(6)$
  • Figure 2: Edge switching (left) and graph stitching (right) operations on $d$-RDR graphs
  • Figure 3: Möbius ladder $\mathop{\mathrm{Ml}}\nolimits(9)$, prism $\mathop{\mathrm{Pr}}\nolimits(12)\cong \mathop{\mathrm{GP}}\nolimits(12,1)$ and Nauru graph $\mathop{\mathrm{GP}}\nolimits(24,5)$ are examples of vertex-transitive 3-RDR graphs of orders $18$ and $24$.
  • Figure 7: 3-RDR colorings of graphs $\mathop{\mathrm{HTG}}\nolimits(3,6,3)$, $\mathop{\mathrm{HTG}}\nolimits(4,6,0)$ and $\mathop{\mathrm{HTG}}\nolimits(7,6,3)$.

Theorems & Definitions (27)

  • Theorem 2.1: Kuzman, kuzman regular
  • Corollary 2.2
  • Example 2.3
  • Proposition 2.4: RDR edge switching
  • proof
  • Example 2.5
  • Proposition 2.6: RDR graph stitching
  • Example 2.7
  • Proposition 3.1
  • Theorem 3.2
  • ...and 17 more