# On Three-Rainbow Dominationof Generalized Petersen Graphs P(ck,k)

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Generalized Petersen Graphs

#### 2.2. Rainbow Domination and Singleton Rainbow Domination

#### 2.3. Graph Covers

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 2.4. Two Constructions

**Construction**

**1.**

- Start with $P(ck,k)$.
- Delete vertices${V}_{c}=\{{v}_{(c-1)k},{v}_{(c-1)k+1},{v}_{(c-1)k+2},\dots {v}_{ck-1}\}$ and${U}_{c}=\{{u}_{(c-1)k},{u}_{(c-1)k+1},{u}_{(c-1)k+2},\dots {u}_{ck-1}\}$and delete all edges incident to these vertices.
- Add edges $\{{v}_{(c-2)k}{v}_{0},{v}_{(c-2)k+1}{v}_{1},{v}_{(c-2)k+2}{v}_{2},\dots {v}_{(c-1)k-1}{v}_{k-1}\}\}$ on the inner cycles and edge $\left\{{u}_{(c-1)k-1}{u}_{0}\right\}$ on the outer cycle.

**Construction**

**2.**

- Start with $P(ck,k)$. Choose $K\in \{0,1,\dots ,k-1\}$. Delete the vertices $Ou{t}_{K}=\left\{{u}_{jk+K}\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}j=0,1,2,\dots ,c-1\}$ and vertices of the corresponding inner cycle $In{n}_{K}=\left\{{v}_{jk+K}\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}j=0,1,2,\dots ,c-1\}$, and delete all edges incident to these vertices.
- Add edges ${u}_{jk+K-1}{u}_{jk+K+1}$ for $j=0,1,2,\dots ,c-1$.

**Proposition**

**3**

**.**Construction 2 on $P(ck,k)$ results in the graph that is isomorphic to $P\left(c\right(k-1),k-1)$.

## 3. Related Previous Work

**Theorem**

**2**

**.**For three-rainbow domination number ${\gamma}_{r3}$ and singleton three-rainbow domination number ${\tilde{\gamma}}_{r3}$ of generalized Petersen graphs $P(6k,k)$ it holds:

- If $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then ${\gamma}_{r3}\left(P(6k,k)\right)={\tilde{\gamma}}_{r3}\left(P(6k,k)\right)=6k$;
- If $k\equiv 0\phantom{\rule{3.33333pt}{0ex}}(mod2)$, then $6k<{\gamma}_{r3}\left(P(6k,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(6k,k)\right)=6k+3$;
- If $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $6k<{\gamma}_{r3}\left(P(6k,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(6k,k)\right)\le 6k+6$.

**Lemma**

**1**

**.**Let $G=P(n,k)$. If ${\gamma}_{r3}\left(G\right)=n=\frac{\left|V\right(G\left)\right|}{2}$, then ${\gamma}_{rt}\left(G\right)={\tilde{\gamma}}_{rt}\left(G\right)$, and any minimal assignment is a singleton $3RD$ function.

**Lemma**

**2**

- Let P be an induced path of length ℓ on vertices $\{{v}_{0},{v}_{1},\dots ,{v}_{\ell}\}$ in G. Assume that one of the vertices ${v}_{0}$ and ${v}_{\ell}$ is uncolored and the other is assigned a color. Then, $w\left(f\left(P\right)\right)\ge \u2308\frac{\ell +1}{2}\u2309$.
- Let C be a cycle of length ℓ. Then, $w\left(f\left(C\right)\right)\ge \u2308\frac{\ell}{2}\u2309$.

## 4. Summary of Our Results

**Theorem**

**3.**

- If $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then ${\gamma}_{r3}\left(P(ck,k)\right)={\tilde{\gamma}}_{r3}\left(P(ck,k)\right)=ck$;
- If $k\equiv 0\phantom{\rule{3.33333pt}{0ex}}(mod2)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)=c(k+\frac{1}{2})$;
- If $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le c(k+1)$.

**Proof.**

**Theorem**

**4.**

- If $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)=ck+\lceil \frac{k}{2}\rceil $;
- If $k\equiv 0\phantom{\rule{3.33333pt}{0ex}}(mod2)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le ck+\lfloor \frac{c}{2}\rfloor +\frac{k}{2}$;
- If $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le c(k+1)+\lceil \frac{k-2}{2}\rceil $.

**Proof.**

**Theorem**

**5.**

- If $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le ck+k+1$;
- If $k\equiv 0\phantom{\rule{3.33333pt}{0ex}}(mod2)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le ck+\frac{c}{2}+k$;
- If $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$, then $ck<{\gamma}_{r3}\left(P(ck,k)\right)\le {\tilde{\gamma}}_{r3}\left(P(ck,k)\right)\le ck+c+k-2$.

**Proof.**

## 5. Proofs

#### 5.1. Case $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$ and $c\equiv 0\phantom{\rule{3.33333pt}{0ex}}(mod6)$

**Proposition**

**4.**

**Proof.**

#### 5.2. Lower Bounds for ${\tilde{\gamma}}_{r3}\left(P(ck,k)\right)$

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Lemma**

**5.**

**Proof.**

- First, assume i is odd. As the neighbors of ${v}_{0}$ on the inner cycle $In{n}_{0}$ are colored by B and G, and since the length of cycle $In{n}_{0}$ is c, for the opposite vertex of ${v}_{0}$, ${v}_{\frac{ck}{2}}={v}_{3ik}$ we have $f\left({v}_{3ik}\right)=R$ because $3ik\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$. Furthermore, the pattern on the outer cycle gives $f\left({u}_{3ik-k}\right)=R$. Hence, vertex ${v}_{3ik-k}$ has two neighbors colored by R, and so f is not a 3RDF. Contradiction.
- Second, assume i is even. In this case, $f\left({u}_{\frac{ck}{2}}\right)=f\left({u}_{3ik}\right)=R$. Then, the two neighbors of ${v}_{3ik}$, vertices ${v}_{3ik-k}$ and ${v}_{3ik+k}$ are not colored by R. Now, consider ${u}_{3ik-k}$. The pattern on the outer cycle implies that its two neighbors, ${u}_{3ik-k-1}$ and ${u}_{3ik-k+1}$, are colored by B and G. Recall that the third neighbor, ${v}_{3ik-k}$, is also not colored by R. Hence, there is no neighbor of ${u}_{3ik-k}$ with color R, and therefore f is not a 3RDF. Again, this is a contradiction.

#### 5.3. Case $k\equiv 1,5\phantom{\rule{3.33333pt}{0ex}}(mod6)$, General c

**Proposition**

**6.**

**Proof.**

_{ck−1}does not have all three colors in the neighborhood. We know that $\mathbb{F}\left({u}_{0}\right)=3$, $\mathbb{F}\left({u}_{ck-2}\right)\ne 2$, and $\mathbb{F}\left({u}_{ck-1}\right)\ne 2$.

**Proposition 7.**

**Proof.**

**Proposition 8.**

**Proof.**

#### 5.4. Case $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$

**Proposition 9.**

**Proof.**

#### 5.5. Case k Even

**Lemma 6.**

**Proof.**

**Lemma 7.**

**Proof.**

**Lemma 8.**

**Proof.**

**Proposition 10.**

**Proposition 11.**

**Proof.**

**Proposition 12.**

**Proof.**

## 6. Conclusions and Ideas for Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A generalized Petersen graph $P(n,k)$ (

**left**) and another way of drawing $P(ck,k)$ (

**right**).

**Figure 2.**Case c odd and $k\equiv 5\phantom{\rule{3.33333pt}{0ex}}(mod6)$. The outer cycle vertices, U, of $P(Ck,k)$ and construction of $P\left(\right(C-1)k,k)$ are $C=6i$. We emphasize the vertices of U that are deleted (one row).

**Figure 3.**The outer cycle vertices, U, of $P(6k,k)$ for $k=9$ and construction of $P(54,9)$. We emphasize the vertices that are deleted (one row). Coloring of some vertices on inner cycles is indicated.

**Figure 4.**The outer cycle vertices, U, of $P(Ck,k)$ and construction of $P(ck,k)$. We emphasize the vertices that are deleted (one row), $c\equiv 5\phantom{\rule{3.33333pt}{0ex}}(mod6)$ and $k\equiv 3\phantom{\rule{3.33333pt}{0ex}}(mod6)$ and the row of vertices whose neighbors are possibly altered.

$f\left({u}_{0}\right)$ | $f\left({u}_{1}\right)$ | ... | $f\left({u}_{i}\right)$ | ... | $f\left({u}_{k-1}\right)$ | $f\left({u}_{k}\right)$ | $f\left({u}_{k+1}\right)$ | ... |

$f\left({u}_{k}\right)$ | $f\left({u}_{k+1}\right)$ | ... | $f\left({u}_{k+i}\right)$ | ... | $f\left({u}_{2k-1}\right)$ | $f\left({u}_{2k}\right)$ | $f\left({u}_{2k+1}\right)$ | ... |

$f\left({u}_{2k}\right)$ | $f\left({u}_{2k+1}\right)$ | ... | $f\left({u}_{2k+i}\right)$ | ... | $f\left({u}_{3k-1}\right)$ | $f\left({u}_{3k}\right)$ | $f\left({u}_{3k+1}\right)$ | ... |

$f\left({u}_{3k}\right)$ | $f\left({u}_{3k+1}\right)$ | ... | $f\left({u}_{3k+i}\right)$ | ... | $f\left({u}_{4k-1}\right)$ | $f\left({u}_{4k}\right)$ | $f\left({u}_{4k+1}\right)$ | ... |

$f\left({u}_{4k}\right)$ | $f\left({u}_{4k+1}\right)$ | ... | $f\left({u}_{4k+i}\right)$ | ... | $f\left({u}_{5k-1}\right)$ | $f\left({u}_{5k}\right)$ | $f\left({u}_{5k+1}\right)$ | ... |

$f\left({u}_{5k}\right)$ | $f\left({u}_{5k+1}\right)$ | ... | $f\left({u}_{5k+i}\right)$ | ... | $f\left({u}_{6k-1}\right)$ | $f\left({u}_{6k}\right)=f\left({u}_{0}\right)$ | $f\left({u}_{1}\right)$ | ... |

0 | 3 | 0 | 2 | 0 | 1 | 0 |

1 | 0 | 3 | 0 | 2 | 0 | 1 |

0 | 1 | 0 | 3 | 0 | 2 | 0 |

2 | 0 | 1 | 0 | 3 | 0 | 2 |

0 | 2 | 0 | 1 | 0 | 3 | 0 |

3 | 0 | 2 | 0 | 1 | 0 | 3 |

0 | 3 | 0 | 2 | 0 |

1 | 0 | 3 | 0 | 2 |

0 | 1 | 0 | 3 | 0 |

2 | 0 | 1 | 0 | 3 |

0 | 2 | 0 | 1 | 0 |

3 | 0 | 2 | 0 | 1 |

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**MDPI and ACS Style**

Rupnik Poklukar, D.; Žerovnik, J.
On Three-Rainbow Dominationof Generalized Petersen Graphs *P*(*ck*,*k*). *Symmetry* **2022**, *14*, 2086.
https://doi.org/10.3390/sym14102086

**AMA Style**

Rupnik Poklukar D, Žerovnik J.
On Three-Rainbow Dominationof Generalized Petersen Graphs *P*(*ck*,*k*). *Symmetry*. 2022; 14(10):2086.
https://doi.org/10.3390/sym14102086

**Chicago/Turabian Style**

Rupnik Poklukar, Darja, and Janez Žerovnik.
2022. "On Three-Rainbow Dominationof Generalized Petersen Graphs *P*(*ck*,*k*)" *Symmetry* 14, no. 10: 2086.
https://doi.org/10.3390/sym14102086