# On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph

## Abstract

**:**

## 1. Introduction

## 2. Cyclic Permutations and Corresponding Configurations of Subgraphs

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

**Lemma**

**1.**

**Proof of**

**Lemma 1.**

**Lemma**

**2.**

**Theorem**

**1.**

**Proof of**

**Theorem 1.**

**Case 1:**${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=0$. Without loss of generality, we can consider the drawing of ${G}^{\ast}$ with the vertex notation like that in Figure 1a. It is obvious that the set ${R}_{D}$ is empty; that is, $r=0$. Thus, we deal with only the configurations belonging to the nonempty set ${\mathcal{M}}_{D}$ and we discuss over all cardinalities of the set ${\mathcal{M}}_{D}$ in the following subcases:

- i.
- $\left|{\mathcal{M}}_{D}\right|\ge 3$. We consider two subcases. Let us first assume that $\{{\mathcal{A}}_{i},{\mathcal{A}}_{j},{\mathcal{A}}_{k}\}\subseteq {\mathcal{M}}_{D}$ with $i+2\equiv j+1\equiv k\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}5\right)$. Without lost of generality, let us consider three different subgraphs ${T}^{n-2},\phantom{\rule{0.277778em}{0ex}}{T}^{n-1},\phantom{\rule{0.277778em}{0ex}}{T}^{n}\in {S}_{D}$ such that ${F}^{n-2}$, ${F}^{n-1}$ and ${F}^{n}$ have configurations ${\mathcal{A}}_{i}$, ${\mathcal{A}}_{j}$, and ${\mathcal{A}}_{k}$, respectively. Then, ${\mathrm{cr}}_{D}({T}^{n-2}\cup {T}^{n-1}\cup {T}^{n},{T}^{m})\ge 14$ holds for any ${T}^{m}\in {S}_{D}$ with $m\ne n-2,n-1,n$ by summing the values in all columns in the considered three rows of Table 1. Moreover, ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n-2}\cup {T}^{n-1}\cup {T}^{n},{T}^{m})\ge 6$ is fulfilling for any subgraph ${T}^{m}\notin {S}_{D}$ by Lemma 1. ${\mathrm{cr}}_{D}({T}^{n-2}\cup {T}^{n-1}\cup {T}^{n})\ge 13$ holds by summing of three corresponding values of Table 1 between the considered configurations ${\mathcal{A}}_{i}$, ${\mathcal{A}}_{j}$, and ${\mathcal{A}}_{k}$, by fixing the subgraph ${G}^{\ast}\cup {T}^{n-2}\cup {T}^{n-1}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{6,n-3}\right)+{\mathrm{cr}}_{D}({K}_{6,n-3},{G}^{\ast}\cup {T}^{n-2}\cup {T}^{n-1}\cup {T}^{n})+{\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n-2}\cup {T}^{n-1}\cup {T}^{n})\\ \ge 6\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+15(s-3)+6(n-s)+13+3=6\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+6n+9s-29\\ \ge 6\u230a\frac{n-3}{2}\u230b\u230a\frac{n-4}{2}\u230b+6n+9\left(\u2308\frac{n}{2}\u2309+1\right)-29\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{6,n-2}\right)+{\mathrm{cr}}_{D}({K}_{6,n-2},{G}^{\ast}\cup {T}^{n-1}\cup {T}^{n})+{\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n-1}\cup {T}^{n})\\ \ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+11(s-2)+4(n-s)+4+2=6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+7s-16\\ \ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+7\left(\u2308\frac{n}{2}\u2309+1\right)-16\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$
- ii.
- $\left|{\mathcal{M}}_{D}\right|=2$; that is, ${\mathcal{M}}_{D}=\{{\mathcal{A}}_{i},{\mathcal{A}}_{j}\}$ for some $i,j\in \{1,\dots ,5\}$ with $i\ne j$. Without lost of generality, let us consider two different subgraphs ${T}^{n-1},\phantom{\rule{0.277778em}{0ex}}{T}^{n}\in {S}_{D}$ such that ${F}^{n-1}$ and ${F}^{n}$ have mentioned configurations ${\mathcal{A}}_{i}$ and ${\mathcal{A}}_{j}$, respectively. Then, ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n-1}\cup {T}^{n},{T}^{m})\ge 1+10=11$ holds for any ${T}^{m}\in {S}_{D}$ with $m\ne n-1,n$ also by Table 1. Thus, by fixing the subgraph ${G}^{\ast}\cup {T}^{n-1}\cup {T}^{n}$, we are able to use the same inequalities as in the previous subcase.
- iii.
- $\left|{\mathcal{M}}_{D}\right|=1$; that is, ${\mathcal{M}}_{D}=\left\{{\mathcal{A}}_{j}\right\}$ for only one $j\in \{1,\dots ,5\}$. Without lost of generality, let us assume that ${T}^{n}\in {S}_{D}$ with the configuration ${\mathcal{A}}_{j}\in {\mathcal{M}}_{D}$ of the subgraph ${F}^{n}$. As ${\mathcal{M}}_{D}=\left\{{\mathcal{A}}_{j}\right\}$, we have ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n},{T}^{k})\ge 1+6=7$ for any ${T}^{k}\in {S}_{D}$, $k\ne n$ provided that ${\mathrm{rot}}_{D}\left({t}_{n}\right)={\mathrm{rot}}_{D}\left({t}_{k}\right)$, for more see [13]. Hence, by fixing the subgraph ${G}^{\ast}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})={\mathrm{cr}}_{D}\left({K}_{6,n-1}\right)+{\mathrm{cr}}_{D}({K}_{6,n-1},{G}^{\ast}\cup {T}^{n})+{\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n})\\ \ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+7(s-1)+3(n-s)+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3n+4s-6\\ \ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3n+4\left(\u2308\frac{n}{2}\u2309+1\right)-6\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$

**Case 2:**${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=1$ with ${\mathrm{cr}}_{D}\left({C}_{6}\left({G}^{\ast}\right)\right)=0$. At first, without loss of generality, we can consider the drawing of ${G}^{\ast}$ with the vertex notation like that in Figure 1b. Since the set ${R}_{D}$ can be nonempty, two possible subcases may occur:

- i.
- Let ${R}_{D}$ be the nonempty set; that is, there is a subgraph ${T}^{i}\in {R}_{D}$. Now, for a ${T}^{i}\in {R}_{D}$, the reader can easily see that the subgraph ${F}^{i}={G}^{\ast}\cup {T}^{i}$ is uniquely represented by ${\mathrm{rot}}_{D}\left({t}_{i}\right)=\left(165432\right)$, and ${\mathrm{cr}}_{D}({T}^{i},{T}^{j})\ge 6$ for any ${T}^{j}\in {R}_{D}$ with $j\ne i$ provided that ${\mathrm{rot}}_{D}\left({t}_{i}\right)={\mathrm{rot}}_{D}\left({t}_{j}\right)$; for more see [13]. Moreover, it is not difficult to verify by a discussion over all possible drawings D that ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{i},{T}^{k})\ge 5$ holds for any subgraph ${T}^{k}\in {S}_{D}$, and ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{i},{T}^{k})\ge 4$ is also fulfilling for any subgraph ${T}^{k}\notin {R}_{D}\cup {S}_{D}$. Thus, by fixing the subgraph ${G}^{\ast}\cup {T}^{i}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+6(r-1)+5s+4(n-r-s)+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b\\ +4n+(2r+s)-5\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+\left(n-\u230a\frac{n}{2}\u230b+1\right)-5\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$
- ii.
- Let ${R}_{D}$ be the empty set; that is, each subgraph ${T}^{j}$ crosses the edges of ${G}^{\ast}$ at least once in D. Thus, we deal with the configurations belonging to the nonempty set ${\mathcal{N}}_{D}$. Let us consider a subgraph ${T}^{j}\in {S}_{D}$ with the configuration ${\mathcal{B}}_{i}\in {\mathcal{N}}_{D}$ of ${F}^{j}$, where $i\in \{1,2,3,4\}$. Then, the lower-bounds of number of crossings of two configurations from $\mathcal{N}$ confirm that ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{j},{T}^{k})\ge 1+4=5$ holds for any ${T}^{k}\in {S}_{D}$, $k\ne j$. Moreover, one can also easily verify over all possible drawings D that ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{j},{T}^{k})\ge 4$ is true for any subgraph ${T}^{k}\notin {S}_{D}$. Hence, by fixing the subgraph ${G}^{\ast}\cup {T}^{j}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+5(s-1)+4(n-s)+1+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b\hfill \\ +4n+s-3\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+\left(\u2308\frac{n}{2}\u2309+1\right)-3\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\hfill \end{array}$$

- i.
- ${\mathcal{E}}_{4}\in {\mathcal{O}}_{D}$. Without lost of generality, let us assume that ${T}^{n}\in {S}_{D}$ with the configuration ${\mathcal{E}}_{4}\in {\mathcal{O}}_{D}$ of ${F}^{n}$. Only for this subcase, one can easily verify over all possible drawings D for which ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n},{T}^{k})\ge 4$ is true for any subgraph ${T}^{k}\notin {S}_{D}$. Thus, by fixing the subgraph ${G}^{\ast}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+5(s-1)+4(n-s)+1+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b\hfill \\ +4n+s-3\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+4n+\left(\u2308\frac{n}{2}\u2309+1\right)-3\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\hfill \end{array}$$
- ii.
- ${\mathcal{E}}_{4}\notin {\mathcal{O}}_{D}$ and ${\mathcal{E}}_{3}\in {\mathcal{O}}_{D}$. Without lost of generality, let us assume that ${T}^{n}\in {S}_{D}$ with the configuration ${\mathcal{E}}_{3}\in {\mathcal{O}}_{D}$ of ${F}^{n}$. In this subcase, ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n},{T}^{k})\ge 1+5=6$ holds for any subgraph ${T}^{k}\in {S}_{D}$, $k\ne n$ by the remaining values in the third row of Table 2. Hence, by fixing the subgraph ${G}^{\ast}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+6(s-1)+3(n-s)+1+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b\\ +3n+3s-4\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3n+3\left(\u2308\frac{n}{2}\u2309+1\right)-4\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$
- iii.
- ${\mathcal{O}}_{D}=\{{\mathcal{E}}_{1},{\mathcal{E}}_{2}\}$. Without lost of generality, let us consider two different subgraphs ${T}^{n-1},\phantom{\rule{0.277778em}{0ex}}{T}^{n}\in {S}_{D}$ such that ${F}^{n-1}$ and ${F}^{n}$ have mentioned configurations ${\mathcal{E}}_{1}$ and ${\mathcal{E}}_{2}$, respectively. Then, ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n-1}\cup {T}^{n},{T}^{k})\ge 1+10=11$ holds for any ${T}^{k}\in {S}_{D}$ with $k\ne n-1,n$ also by Table 2. Thus, by fixing the subgraph ${G}^{\ast}\cup {T}^{n-1}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+11(s-2)+4(n-s)+4+2=6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b\\ +4n+7s-16\ge 6\u230a\frac{n-2}{2}\u230b\u230a\frac{n-3}{2}\u230b+4n+7\left(\u2308\frac{n}{2}\u2309+1\right)-16\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$
- iv.
- ${\mathcal{O}}_{D}=\left\{{\mathcal{E}}_{i}\right\}$ for only one $i\in \{1,2\}$. Without lost of generality, let us assume that ${T}^{n}\in {S}_{D}$ with the configuration ${\mathcal{E}}_{1}$ of ${F}^{n}$. In this subcase, ${\mathrm{cr}}_{D}({G}^{\ast}\cup {T}^{n},{T}^{k})\ge 1+6=7$ holds for any ${T}^{k}\in {S}_{D}$, $k\ne n$ provided that ${\mathrm{rot}}_{D}\left({t}_{n}\right)={\mathrm{rot}}_{D}\left({t}_{k}\right)$. Hence, by fixing the subgraph ${G}^{\ast}\cup {T}^{n}$,$$\begin{array}{c}{\mathrm{cr}}_{D}({G}^{\ast}+{D}_{n})\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+7(s-1)+3(n-s)+1=6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b\\ +3n+4s-6\ge 6\u230a\frac{n-1}{2}\u230b\u230a\frac{n-2}{2}\u230b+3n+4\left(\u2308\frac{n}{2}\u2309+1\right)-6\ge 6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b.\end{array}$$

**Case 3:**${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=2$ with ${\mathrm{cr}}_{D}\left({C}_{6}\left({G}^{\ast}\right)\right)=0$. At first, without loss of generality, we can consider the drawing of ${G}^{\ast}$ with the vertex notation like that in Figure 1c. It is obvious that the set ${R}_{D}$ is empty, that is, the set ${S}_{D}$ cannot be empty. Our aim is to list again all possible rotations ${\mathrm{rot}}_{D}\left({t}_{j}\right)$ which can appear in D if a subgraph ${T}^{j}\in {S}_{D}$. Since there is only one subdrawing of ${F}^{j}\setminus \left\{{v}_{1}\right\}$ represented by the rotation $\left(26543\right)$, there are three ways to obtain the subdrawing of ${F}^{j}$ depending on which edge of ${G}^{\ast}$ is crossed by the edge ${t}_{j}{v}_{1}$. These three possible ways under our consideration can be denoted by ${\mathcal{C}}_{k}$, for $k=1,2,3$. Based on the aforementioned arguments, we assume the drawings shown in Figure 7.

**Case 4:**${\mathrm{cr}}_{D}\left({G}^{\ast}\right)\ge 1$ with ${\mathrm{cr}}_{D}\left({C}_{6}\left({G}^{\ast}\right)\right)\ge 1$. For all possible subdrawings of the graph ${G}^{\ast}$ with at least one crossing among edges of ${C}_{6}\left({G}^{\ast}\right)$, and also with the possibility of obtaining a subgraph ${T}^{j}$ that crosses the edges of ${G}^{\ast}$ at most once, one of the ideas of the previous subcases can be applied.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Garey, M.R.; Johnson, D.S. Crossing number is NP-complete. SIAM J. Algebraic. Discret. Methods
**1983**, 4, 312–316. [Google Scholar] [CrossRef] - Klešč, M. The join of graphs and crossing numbers. Electron. Notes Discret. Math.
**2007**, 28, 349–355. [Google Scholar] [CrossRef] - Kulli, V.R.; Muddebihal, M.H. Characterization of join graphs with crossing number zero. Far East J. Appl. Math.
**2001**, 5, 87–97. [Google Scholar] - Kleitman, D.J. The crossing number of K
_{5,n}. J. Comb. Theory**1970**, 9, 315–323. [Google Scholar] [CrossRef] [Green Version] - Klešč, M.; Schrötter, Š. The crossing numbers of join products of paths with graphs of order four. Discuss. Math. Graph Theory
**2011**, 31, 312–331. [Google Scholar] [CrossRef] - Berežný, Š.; Staš, M. Cyclic permutations and crossing numbers of join products of symmetric graph of order six. Carpathian J. Math.
**2018**, 34, 143–155. [Google Scholar] - Klešč, M. The crossing numbers of join of the special graph on six vertices with path and cycle. Discret. Math.
**2010**, 310, 1475–1481. [Google Scholar] [CrossRef] [Green Version] - Staš, M. Cyclic permutations: Crossing numbers of the join products of graphs. In Proceedings of the Aplimat 2018: 17th Conference on Applied Mathematics, Bratislava, Slovakia, 6–8 February 2018; pp. 979–987. [Google Scholar]
- Staš, M. Determining crossing numbers of graphs of order six using cyclic permutations. Bull. Aust. Math. Soc.
**2018**, 98, 353–362. [Google Scholar] [CrossRef] - Hernández-Vélez, C.; Medina, C.; Salazar, G. The optimal drawing of K
_{5,n}. Electron. J. Comb.**2014**, 21, 29. [Google Scholar] - Berežný, Š.; Buša, J., Jr.; Staš, M. Software solution of the algorithm of the cyclic-order graph. Acta Electrotech. Inform.
**2018**, 18, 3–10. [Google Scholar] [CrossRef] - Klešč, M.; Schrötter, Š. The crossing numbers of join of paths and cycles with two graphs of order five. In Lecture Notes in Computer Science: Mathematical Modeling and Computational Science; Springer: Berlin/Heidelberg, Germany, 2012; Volume 7125, pp. 160–167. [Google Scholar]
- Woodall, D.R. Cyclic-order graphs and Zarankiewicz’s crossing number conjecture. J. Graph Theory
**1993**, 17, 657–671. [Google Scholar] [CrossRef] - Chimani, M.; Wiedera, T. An ILP-based proof system for the crossing number problem. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA 2016), Aarhus, Denmark, 22–24 August 2016; Volume 29, pp. 1–13. [Google Scholar]

**Figure 1.**Six possible drawings of ${G}^{\ast}$ with no crossing among edges of ${C}_{6}\left({G}^{\ast}\right)$. (

**a**): the planar drawing of ${G}^{\ast}$; (

**b**): the drawing of ${G}^{\ast}$ with ${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=1$ and without crossing on edges of ${C}_{6}\left({G}^{\ast}\right)$; (

**c**): the drawing of ${G}^{\ast}$ only with two crossings on edges of ${C}_{6}\left({G}^{\ast}\right)$; (

**d**): the drawing of ${G}^{\ast}$ with ${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=2$ and with one crossing on edges of ${C}_{6}\left({G}^{\ast}\right)$; (

**e**): the drawing of ${G}^{\ast}$ only with one crossing on edges of ${C}_{6}\left({G}^{\ast}\right)$; (

**f**): the drawing of ${G}^{\ast}$ with ${\mathrm{cr}}_{D}\left({G}^{\ast}\right)=2$ and with one crossing on edges of ${C}_{6}\left({G}^{\ast}\right)$.

**Figure 6.**The good drawing of ${G}^{\ast}+{D}_{n}$ with $6\u230a\frac{n}{2}\u230b\u230a\frac{n-1}{2}\u230b+n+\u230a\frac{n}{2}\u230b$ crossings.

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

- | ${\mathcal{A}}_{1}$ | ${\mathcal{A}}_{2}$ | ${\mathcal{A}}_{3}$ | ${\mathcal{A}}_{4}$ | ${\mathcal{A}}_{5}$ |
---|---|---|---|---|---|

${\mathcal{A}}_{1}$ | 6 | 4 | 5 | 5 | 4 |

${\mathcal{A}}_{2}$ | 4 | 6 | 4 | 5 | 5 |

${\mathcal{A}}_{3}$ | 5 | 4 | 6 | 4 | 5 |

${\mathcal{A}}_{4}$ | 5 | 5 | 4 | 6 | 4 |

${\mathcal{A}}_{5}$ | 4 | 5 | 5 | 4 | 6 |

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

- | ${\mathcal{E}}_{1}$ | ${\mathcal{E}}_{2}$ | ${\mathcal{E}}_{3}$ | ${\mathcal{E}}_{4}$ |
---|---|---|---|---|

${\mathcal{E}}_{1}$ | 6 | 4 | 5 | 4 |

${\mathcal{E}}_{2}$ | 4 | 6 | 5 | 5 |

${\mathcal{E}}_{3}$ | 5 | 5 | 6 | 4 |

${\mathcal{E}}_{4}$ | 4 | 5 | 4 | 6 |

**Table 3.**The necessary number of crossings between ${T}^{i}$ and ${T}^{j}$ for the configurations ${\mathcal{F}}_{k}$ and ${\mathcal{F}}_{l}$.

- | ${\mathcal{F}}_{1}$ | ${\mathcal{F}}_{2}$ | ${\mathcal{F}}_{3}$ |
---|---|---|---|

${\mathcal{F}}_{1}$ | 6 | 4 | 5 |

${\mathcal{F}}_{2}$ | 4 | 6 | 5 |

${\mathcal{F}}_{3}$ | 5 | 5 | 6 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Staš, M.
On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph. *Symmetry* **2020**, *12*, 135.
https://doi.org/10.3390/sym12010135

**AMA Style**

Staš M.
On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph. *Symmetry*. 2020; 12(1):135.
https://doi.org/10.3390/sym12010135

**Chicago/Turabian Style**

Staš, Michal.
2020. "On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph" *Symmetry* 12, no. 1: 135.
https://doi.org/10.3390/sym12010135