# On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles
## 2 Cyclic Permutations and Configurations
The same method as above can be applied to establish the remaining lower bounds of two configurations from \(\mathcal{M}\cup\mathcal{N}\). All resulting lower bounds are summarized in the common symmetric Table 1 in which \(\mathcal{X}_{p}\) and \(\mathcal{Y}_{q}\) are configurations of two different subgraphs \(F^{i}\) and \(F^{j}\), where \(\mathcal{X},\mathcal{Y}\in\{\mathcal{A},\mathcal{B}\}\) and if \(\mathcal{X}=\mathcal{A}\) or \(\mathcal{Y}=\mathcal{A}\) then \(p\in\{1,2,3,4\}\) or \(q\in\{1,2,3,4\}\); otherwise, \(p\in\{1,2\}\) or \(q\in\{1,2\}\), respectively. **3. The Crossing Number of \(G^{*}+D_{n}\)**  
In a good drawing of \(G^{*}+D_{n}\), two different vertices \(t_{i}\) and \(t_{j}\) of the graph \(G^{*}+D_{n}\) are said to be _antipodal_ if the edges of the corresponding subgraph \(T^{i}\cup T^{j}\) do not cross each other. A drawing is said to be _antipode-free_ if it does not contain any two antipodal vertices. **Lemma 2**. : _For \(n>2\), let \(D\) be a good and antipode-free drawing of \(G^{*}+D_{n}\) with the subdrawing of \(G^{*}\) induced by \(D\) given in Figure 1a. For some \(p\in\{1,2\}\), if there is a subgraph \(T^{i}\in R_{D}\) with the configuration \(\mathcal{A}_{p}\in\mathcal{M}_{D}\) of \(F^{i}\) and a subgraph \(T^{k}\in S_{D}\) such that \(\operatorname{cr}_{D}(T^{i},T^{k})=2\), then_  
_(a)_ \(\operatorname{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{l})\geq 8\) _for any subgraph_ \(T^{l}\in R_{D}\)_,_ \(l\
eq i\)_; and_  
_(b)_ \(\operatorname{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{l})\geq 7\) _for any subgraph_ \(T^{l}\in S_{D}\)_,_ \(l\
eq k\)_; and_  
_(c)_ \(\operatorname{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{l})\geq 6\) _for any subgraph_ \(T^{l}\
ot\in R_{D}\cup S_{D}\)_._  
Proof. : Let \(\mathcal{A}_{1}\) be the configuration of \(F^{i}\) and remark that it is uniquely represented by its \(\operatorname{rot}_{D}(t_{i})=(123654)\). The induced subdrawing \(D(G^{*}\cup T^{i})\) of \(F^{i}\) contains just six regions with the vertex \(t_{i}\) on their boundaries. If we consider a subgraph \(T^{k}\in S_{D}\) satisfying the restriction \(\operatorname{cr}_{D}(T^{i},T^{k})=2\), then the corresponding vertex \(t_{k}\) can only be placed in the region with the four vertices \(v_{1}\), \(v_{2}\), \(v_{3}\), and \(v_{4}\) of \(G^{*}\) on its boundary. Using this knowledge, the edges \(t_{k}v_{1}\), \(t_{k}v_{3}\), and \(t_{k}v_{4}\) produce no crossings on edges of \(F^{i}\), and the edge \(t_{k}v_{2}\) either crosses the edge \(t_{i}v_{1}\) or does not cross any edge of \(F^{i}\). If \(t_{k}v_{2}\) crosses \(t_{i}v_{1}\), then \(\operatorname{rot}_{D}(t_{k})=(125463)\). If the edges of \(F^{i}\) are not crossed by \(t_{k}v_{2}\) and also \(t_{k}v_{5}\) crosses \(v_{3}v_{4}\) of \(G^{*}\), then \(\operatorname{rot}_{D}(t_{k})=(164532)\). Finally, if \(t_{k}v_{5}\) crosses \(t_{i}v_{4}\), then \(t_{k}v_{6}\) crosses either \(v_{2}v_{3}\), \(t_{i}v_{3}\) with \(\operatorname{rot}_{D}(t_{k})=(154362)\) or \(v_{4}v_{5}\), \(t_{i}v_{4}\) with \(\operatorname{rot}_{D}(t_{k})=(156432)\). In the following, we suppose \(T^{k}\in S_{D}\) with \(\operatorname{rot}_{D}(t_{k})=(125463)\), but a similar idea can be applied to the other three cases mentioned above. 1. Let us assume \(T^{l}\in R_{D}\), \(l\
eq i\), with the configuration \(\mathcal{A}_{p}\in\mathcal{M}_{D}\) of \(F^{l}\) for some \(p\in\{1,\dots,4\}\). Table 2 summarizes the minimal values of necessary crossings among the edges in the subdrawing \(D(T^{i}\cup T^{k}\cup T^{l})\).