# On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles
## 6 The Crossing Numbers of Join Products of Cycles with Six Graphs of Order Six
Five vertices \(v_{2}\), \(v_{3}\), \(v_{4}\), \(v_{5}\), and \(v_{6}\) of \(G^{*}\) are placed in one region of \(D(C_{n}^{*})\) and any subgraph \(T^{v_{i}}\), \(i=2,\ldots,6\), cross no edge of \(G^{*}\). We obtain a contradiction in \(D\) using the same estimate as in the previous case. **Case 3:** Some edge of \(C_{n}^{*}\) crosses the edge \(v_{2}v_{3}\) of \(G^{*}\). Again by Lemma 5, there exist at least  
\[\left(\frac{4}{2}\right)\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac {n-1}{2}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n -1}{2}\right\rfloor+1+\operatorname{cr}_{D}(G^{*})+s+2(n-r-s) \tag{9}\]  
crossings in \(D\), where \(|R_{D}|=r\) and \(|S_{D}|=s\). The empty set \(R_{D}\) contradicts the assumption of \(D\) for all \(n\geq 3\) and we also obtain at least \(6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor+2 \left\lfloor\frac{n}{2}\right\rfloor+2\) crossings in the obtained number (9) for all integers \(n\) more than \(6\). For \(3\leq n\leq 6\) and \(r\geq 1\), we suppose only two subdrawings of \(G^{*}\) presented in Figure 1. Let us first consider the planar subdrawing of \(G^{*}\) induced by \(D\) given in Figure 1a. For \(n=3\), if either \(r=2\), \(s=1\) or \(r=3\), we obtain at least \(10\) and \(12\) crossings in \(D\) using only some of the values in Table 1, respectively. The similar idea can be applied for \(n=4\) if \(r\geq 3\). For \(r=2\) and \(s=2\), if we would like to get the smallest possible number of crossings equal to \(17\) over the four subgraphs with configurations \(\mathcal{A}_{1}\), \(\mathcal{A}_{2}\), \(\mathcal{B}_{1}\), and \(\mathcal{B}_{2}\) (in an effort to obtain their lower bounds in Table 1), we obtain two additional crossings on the edge of \(C_{n}^{*}\) by which the edge \(v_{2}v_{3}\) of \(G^{*}\) is crossed. For \(n=5\) and \(n=6\), it is sufficient, due to (9), to deal only with cases \(n=r\), which again, only thanks to the values from Table 1, give us the numbers of crossings contradicting the assumption in \(D\). Finally, assume the nonplanar subdrawing of \(G^{*}\) induced by \(D\) given in Figure 1b. The number of crossings obtained in (9) confirms a contradiction in \(D\) for all \(n\) at least \(5\). For \(n=3\) and \(n=4\), if either \(n=r\) or \(n=4\), \(r=3\), \(s=1\), we can verify the contradicting numbers of crossings in \(D\) by a very fast recalculation because the edges of \(T^{i}\cup T^{j}\) and \(T^{i}\cup T^{k}\) cross each other at least five and three times for any three different subgraphs \(T^{i},T^{j}\in R_{D}\), \(T^{k}\in S_{D}\), respectively. The proof of Theorem 6 is done. \(\qed\)  
In both Figures 6 and 7 by adding the edge \(v_{4}v_{6}\), it is possible to add the edge \(t_{1}t_{n}\) that creates \(C_{n}^{*}\) on the vertices of \(P_{n}^{*}\) with just two additional crossings. Thus, the result of Corollary 6 is obvious, because \(G^{*}+C_{n}\) is a subgraph of \(G_{2}+C_{n}\), and so \(\operatorname{cr}(G_{2}+C_{n})\geq\operatorname{cr}(G^{*}+C_{n})\). **Corollary 6**. : \(\operatorname{cr}(G_{2}+C_{n})=6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor \frac{n-1}{2}\right\rfloor+2\left\lfloor\frac{n}{2}\right\rfloor+2\) _for \(n\geq 3\)._  
**Theorem 7**. : \(\operatorname{cr}(H^{*}+C_{n})=6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor \frac{n-1}{2}\right\rfloor+2\left\lfloor\frac{n}{2}\right\rfloor+2\) _for \(n\geq 3\)._  
Proof. : The proof proceeds similarly like for the graph \(G^{*}+C_{n}\) in Theorem 6. In Figure 9, adding the edge \(t_{1}t_{n}\) to \(P_{n}^{*}\) offers the good drawing of \(H^{*}+C_{n}\) with \(6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor+2 \left\lfloor\frac{n}{2}\right\rfloor+2\) crossings. If we assume a drawing \(D\) of \(H^{*}+C_{n}\) with at most \(6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor+2 \left\lfloor\frac{n}{2}\right\rfloor+1\) crossings, then the result of Theorem 2 forces at most one crossing on the edges of \(C_{n}^{*}\) in \(D\).