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On rainbow domination of cubic graphs

Janez Žerovnik

Abstract

The structure of minimal weight rainbow domination functions of cubic graphs are studied. Based on general observations for cubic graphs, generalized Petersen graphs $P(ck,k)$ are characterized whose 4- and 5-rainbow domination numbers equal the general lower bounds. As $t$-rainbow domination of cubic graphs for $t \ge 6$ is trivial, characterizations of such generalized Petersen graphs $P(ck,k)$ are known for all $t$-rainbow domination numbers.In addition, new upper bounds for 4- and 5-rainbow domination numbers that are valid for all $P(ck,k)$ are provided.

On rainbow domination of cubic graphs

Abstract

The structure of minimal weight rainbow domination functions of cubic graphs are studied. Based on general observations for cubic graphs, generalized Petersen graphs are characterized whose 4- and 5-rainbow domination numbers equal the general lower bounds. As -rainbow domination of cubic graphs for is trivial, characterizations of such generalized Petersen graphs are known for all -rainbow domination numbers.In addition, new upper bounds for 4- and 5-rainbow domination numbers that are valid for all are provided.
Paper Structure (11 sections, 29 theorems, 14 equations, 1 figure)

This paper contains 11 sections, 29 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Previous Work
  3. Bounds for rainbow domination of cubic graphs
  4. Lower bounds for 3-regular graphs
  5. General upper bound
  6. Rainbow domination of Petersen graphs
  7. More definitions and previous work
  8. New upper bounds for $\gamma_{r4}(P(n,k))$ and $\gamma_{r5}(P(n,k))$.
  9. Improved upper bounds
  10. Lower bounds for $\gamma_{r4}(P(n,k))$ and $\gamma_{r5}(P(n,k))$ - extremal examples.
  11. Conclusions

Key Result

Theorem 1

Let $t\in\{3,4,5\}$. Then $\gamma_{rt}(P(n, k)) = \frac{t}{3}n$ if and only if $n\equiv 0~\pmod 6$, and $k \equiv 1, 5~\pmod 6$.

Figures (1)

  • Figure 1: A generalized Petersen graph $P(n,k)$

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Corollary 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Lemma 8
  • ...and 31 more