# Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

## Abstract

**:**

## 1. Introduction

## 2. Cyclic Permutations and Configurations

## 3. The Graph of Configurations ${\mathcal{G}}_{\mathit{D}}$

## 4. The Crossing Number of $\mathit{G}+{\mathit{D}}_{\mathit{n}}$

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

- i.
- $\{{A}_{1},{A}_{2}\}\subseteq {\mathcal{M}}_{D}$, i.e., ${w}_{D}\left({a}_{1}{a}_{2}\right)=1$. Without loss of generality, let us consider two different subgraphs ${T}^{n},\phantom{\rule{0.277778em}{0ex}}{T}^{n-1}\in {R}_{D}$ such that ${F}^{n}$ and ${F}^{n-1}$ have configurations ${A}_{1}$ and ${A}_{2}$, respectively. Then, ${\mathrm{cr}}_{D}(G\cup {T}^{n}\cup {T}^{n-1},{T}^{i})\ge 5$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n-1,n$ by summing the values in all columns in the considered two rows of Table 1. Moreover, ${\mathrm{cr}}_{D}({T}^{n}\cup {T}^{n-1},{T}^{i})\ge 3$ for any subgraph ${T}^{i}$ with $i\ne n-1,n$ due to the properties of the cyclic permutations. Hence, by fixing the graph $G\cup {T}^{n}\cup {T}^{n-1}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+5(r-2)+4(n-r)+1=4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+r-9$$$$\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+\left(\u2308\frac{n}{2}\u2309+1\right)-9\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$
- ii.
- $\{{A}_{1},{A}_{2}\}\u2288{\mathcal{M}}_{D}$, i.e., ${w}_{D}\left(e\right)\ge 2$ for any $e\in {E}_{D}$.Let us assume that $\{{A}_{1},{B}_{2},{B}_{4}\}\subseteq {\mathcal{M}}_{D}$ or $\{{A}_{2},{B}_{1},{B}_{3}\}\subseteq {\mathcal{M}}_{D}$, i.e., there is a three-cycle in the graph ${\mathcal{G}}_{D}$ with weights of two of all its edges. Without loss of generality, let us consider three different subgraphs ${T}^{n},\phantom{\rule{0.277778em}{0ex}}{T}^{n-1}\phantom{\rule{0.277778em}{0ex}}{T}^{n-2}\in {R}_{D}$ such that ${F}^{n}$, ${F}^{n-1}$m and ${F}^{n-2}$ have different configurations from $\{{A}_{1},{B}_{2},{B}_{4}\}$. Then, ${\mathrm{cr}}_{D}(G\cup {T}^{n}\cup {T}^{n-1}\cup {T}^{n-2},{T}^{i})\ge 8$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n-1,n$ by Table 1, and ${\mathrm{cr}}_{D}(G\cup {T}^{n}\cup {T}^{n-1}\cup {T}^{n-2},{T}^{i})\ge 5$ for any subgraph ${T}^{i}\in {S}_{D}$ by Lemma 1. Thus, by fixing the graph $G\cup {T}^{n}\cup {T}^{n-1}\cup {T}^{n-2}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+8(r-3)+5(n-r)+6\ge 4\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+5n+3r-18$$$$\ge 4\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+5n+3\left(\u2308\frac{n}{2}\u2309+1\right)-18\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$

- (1)
- $\{{A}_{j},{B}_{k}\}\subseteq {\mathcal{M}}_{D}$ for some $k\equiv j+1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}2)$ or $\{{B}_{j},{B}_{j+2}\}\subseteq {\mathcal{M}}_{D}$, where $j\in \{1,2\}$. Without loss of generality, let us consider two different subgraphs ${T}^{n},\phantom{\rule{0.277778em}{0ex}}{T}^{n-1}\in {R}_{D}$ such that ${F}^{n}$ and ${F}^{n-1}$ have configurations ${A}_{1}$ and ${B}_{2}$, respectively. Then, ${\mathrm{cr}}_{D}(G\cup {T}^{n}\cup {T}^{n-1},{T}^{i})\ge 6$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n-1,n$ by Table 1. Moreover, ${\mathrm{cr}}_{D}({T}^{n}\cup {T}^{n-1},{T}^{i})\ge 2$ for any subgraph ${T}^{i}$ with $i\ne n-1,n$ due to properties of the cyclic permutations. Hence, if we fix the graph $G\cup {T}^{n}\cup {T}^{n-1}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+6(r-2)+3s+4(n-r-s)+2=4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b$$$$+4n+r+r-s-10\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+\u2308\frac{n}{2}\u2309+1+1-10\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$
- (2)
- $\{{A}_{j},{B}_{k}\}\u2288{\mathcal{M}}_{D}$ for any $k\equiv j+1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}2)$ and $\{{B}_{j},{B}_{j+2}\}\u2288{\mathcal{M}}_{D}$, where $j=1,2$, i.e., ${w}_{D}\left(e\right)\ge 3$ for any $e\in {E}_{D}$. Without loss of generality, we can assume that ${T}^{n}\in {R}_{D}$. Then, ${\mathrm{cr}}_{D}({T}^{n},{T}^{i})\ge 3$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n$. Thus, by fixing the graph $G\cup {T}^{n}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3(r-1)+2(n-r)+0=4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+2n+r-3$$$$\ge 4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+2n+\left(\u2308\frac{n}{2}\u2309+1\right)-3\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$

- i.
- $\{{A}_{i},{A}_{i+1}\}\subseteq {\mathcal{N}}_{D}$ for some $i\in \{1,2\}$. Without loss of generality, let us consider two different subgraphs ${T}^{n},\phantom{\rule{0.277778em}{0ex}}{T}^{n-1}\in {R}_{D}$ such that ${F}^{n}$ and ${F}^{n-1}$ have different configurations from the set $\{{A}_{1},{A}_{2}\}$. Then, ${\mathrm{cr}}_{D}(G\cup {T}^{n}\cup {T}^{n-1},{T}^{i})\ge 6$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n-1,n$ by Table 2. Moreover, ${\mathrm{cr}}_{D}({T}^{n}\cup {T}^{n-1},{T}^{i})\ge 2$ for any subgraph ${T}^{i}$ with $i\ne n-1,n$ due to the properties of the cyclic permutations. Hence, by fixing the graph $G\cup {T}^{n}\cup {T}^{n-1}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+6(r-2)+3s+4(n-r-s)+2+1=4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b$$$$+4n+r+r-s-9\ge 4\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+\u2308\frac{n}{2}\u2309+1+1-9\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$If ${F}^{n}$ and ${F}^{n-1}$ have different configurations from the set $\{{A}_{3},{A}_{4}\}$, then the same argument can be applied.
- ii.
- $\{{A}_{i},{A}_{i+1}\}\u2288{\mathcal{N}}_{D}$ for any $i=1,2$. Without loss of generality, we can assume that ${T}^{n}\in {R}_{D}$. Then, ${\mathrm{cr}}_{D}({T}^{n},{T}^{i})\ge 3$ for any ${T}^{i}\in {R}_{D}$ with $i\ne n$. Thus, by fixing the graph $G\cup {T}^{n}$,$${\mathrm{cr}}_{D}(G+{D}_{n})\ge 4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3(r-1)+2(n-r)+1=4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+2n+r-2$$$$\ge 4\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+2n+\left(\u2308\frac{n}{2}\u2309+1\right)-2\ge 4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b.$$

## 5. Three Other Graphs

**Corollary**

**1.**

**Theorem**

**2.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two good drawings of the graph G. (

**a**): the planar drawing of G; (

**b**): the drawing of G with cr

_{D}(G) = 1.

**Figure 4.**The good drawings of $G+{D}_{2}$ and of $G+{D}_{n}$. (

**a**): the drawing of G + D

_{2}with one crossing; (

**b**): the drawing of G + D

_{n}with $4\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+\u230a\frac{n}{2}\u230b$ crossings.

**Table 1.**The necessary number of crossings between ${T}^{i}$ and ${T}^{j}$ for the configurations ${X}_{k}$, ${Y}_{l}$.

− | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | ${\mathit{B}}_{4}$ | ${\mathit{B}}_{5}$ | ${\mathit{B}}_{6}$ |
---|---|---|---|---|---|---|---|---|

${A}_{1}$ | 4 | 1 | 3 | 2 | 3 | 2 | 3 | 2 |

${A}_{2}$ | 1 | 4 | 2 | 3 | 2 | 3 | 2 | 3 |

${B}_{1}$ | 3 | 2 | 4 | 3 | 2 | 3 | 4 | 3 |

${B}_{2}$ | 2 | 3 | 3 | 4 | 3 | 2 | 3 | 4 |

${B}_{3}$ | 3 | 2 | 2 | 3 | 4 | 3 | 4 | 3 |

${B}_{4}$ | 2 | 3 | 3 | 2 | 3 | 4 | 3 | 4 |

${B}_{5}$ | 3 | 2 | 4 | 3 | 4 | 3 | 4 | 3 |

${B}_{6}$ | 2 | 3 | 3 | 4 | 3 | 4 | 3 | 4 |

**Table 2.**The necessary number of crossings between ${T}^{i}$ and ${T}^{j}$ for the configurations ${A}_{k}$, ${A}_{l}$.

− | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ |
---|---|---|---|---|

${A}_{1}$ | 4 | 2 | 3 | 3 |

${A}_{2}$ | 2 | 4 | 3 | 3 |

${A}_{3}$ | 3 | 3 | 4 | 2 |

${A}_{4}$ | 3 | 3 | 2 | 4 |

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**MDPI and ACS Style**

Staš, M.
Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five. *Symmetry* **2019**, *11*, 123.
https://doi.org/10.3390/sym11020123

**AMA Style**

Staš M.
Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five. *Symmetry*. 2019; 11(2):123.
https://doi.org/10.3390/sym11020123

**Chicago/Turabian Style**

Staš, Michal.
2019. "Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five" *Symmetry* 11, no. 2: 123.
https://doi.org/10.3390/sym11020123