# On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles
## 2 Cyclic Permutations and Configurations
All these subcases contradict the assumption (5) in \(D\), and therefore, the case of \(|\mathcal{M}_{D}|=2\) offers only two possibilities of either \(\mathcal{M}_{D}=\{\mathcal{A}_{1},\mathcal{A}_{4}\}\) or \(\mathcal{M}_{D}=\{\mathcal{A}_{2},\mathcal{A}_{3}\}\). If we fix any two subgraphs \(T^{i}\), \(T^{j}\in R_{D}\) such that \(F^{i}\), \(F^{j}\) have the configurations \(\mathcal{A}_{1}\) and \(\mathcal{A}_{4}\), respectively, then Table 1 confirms that the condition (3) holds. The condition (4) follows from the special assumption \(\mathcal{N}_{D}=\emptyset\) of Subcase 1. As \(\text{cr}_{D}(T^{i}\cup T^{j})\geq 4\), the discussed drawing again contradicts the assumption of \(D\) by Lemma 3. Finally, if \(\mathcal{M}_{D}=\{\mathcal{A}_{p}\}\) for only one \(p\in\{1,2\}\), the proof can proceed in the same way as in the case of \(\mathcal{A}_{p}\in\mathcal{M}_{D}\) for only \(p\in\{3,4\}\). **Subcase 2:** Let \(\mathcal{N}_{D}\) be a nonempty set, that is, there is a possibility to obtain a subgraph \(T^{k}\in S_{D}\) satisfying \(\text{cr}_{D}(G^{*}\cup T^{i},T^{k})\geq 2\) for some \(T^{i}\in R_{D}\) with either \(\mathcal{A}_{1}\) or \(\mathcal{A}_{2}\) of \(F^{i}\). Let us denote \(S_{D}(\mathcal{B}_{1})=\{T^{k}\in S_{D}:\text{rot}_{D}(t_{k})=(154632)\}\), \(S_{D}(B_{2})=\{T^{k}\in S_{D}:\text{rot}_{D}(t_{k})=(123465)\}\), and consequently \(s_{1}=|S_{D}(\mathcal{B}_{1})|\), \(s_{2}=|S_{D}(B_{2})|\). Notice that \(S_{D}(\mathcal{B}_{1})\) and \(S_{D}(\mathcal{B}_{2})\) are two disjoint subsets of \(S_{D}\), and \(s_{1}+s_{2}\leq s\), that is, \(s-s_{1}-s_{2}\geq 0\). Using their symmetry, let \(s_{1}\) be greater than \(s_{2}\) provided that at least one of the sets \(S_{D}(\mathcal{B}_{1})\) and \(S_{D}(\mathcal{B}_{2})\) is nonempty. Now, we discuss the four possible subcases:  
1. \(\mathcal{A}_{1}\in\mathcal{M}_{D}\) and assume some subgraph \(F^{i}\), having the configuration \(\mathcal{A}_{1}\). Let \(T^{k}\) be a subgraph from the nonempty set \(S_{D}(\mathcal{B}_{1})\). By summing the values in two considered rows for four possible columns of Table 1, we obtain \(\text{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{j})\geq 7\) for any \(T^{j}\in R_{D}\) with \(j\
eq i\). The subgraph \(F^{k}\) is represented by the cyclic permutation \((154632)\), and so \(\text{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{i})\geq 1+1+6=8\) holds for any other \(T^{l}\in S_{D}(\mathcal{B}_{1})\), \(l\
eq k\). Moreover, \(\text{cr}_{D}(T^{i}\cup T^{k},T^{l})\geq 5\) is fulfilling for any \(T^{l}\) with \(l\
eq i,k\) because five interchanges of adjacent elements of \(\text{rot}_{D}(t_{i})\) produce \(\text{rot}_{D}(t_{k})\). This forces \(\text{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{l})\geq 1+5=6\) for any \(T^{l}\in S_{D}(\mathcal{B}_{2})\) and \(\text{cr}_{D}(G^{*}\cup T^{i}\cup T^{k},T^{l})\geq 2+5=7\) for any \(T^{l}\
ot\in R_{D}\cup S_{D}\). As \(\text{cr}_{D}(G^{*}\cup T^{i}\cup T^{k})\geq 2\), by fixing the subgraph \(G^{*}\cup T^{i}\cup T^{k}\), we have \[\operatorname{cr}_{D}(G^{*}+D_{n})=\operatorname{cr}_{D}(K_{6,n})+ \operatorname{cr}_{D}(G^{*})+\operatorname{cr}_{D}(G^{*},K_{6,n})\geq 6 \left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor+n\]4 \[\operatorname{cr}_{D}(G^{*}+D_{n})=\operatorname{cr}_{D}(K_{6,n})+ \operatorname{cr}_{D}(G^{*})+\operatorname{cr}_{D}(G^{*},K_{6,n})\geq 6 \left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor+n\]5
2. \(\mathcal{A}_{1}\
ot\in\mathcal{M}_{D}\) and \(\mathcal{A}_{3}\in\mathcal{M}_{D}\). Let us assume the configuration \(\mathcal{A}_{3}\) of some subgraph \(F^{i}\), and let \(T^{k}\in S_{D}(\mathcal{B}_{1})\). Taking into account the subgraph \(G^{*}\cup T^{i}\cup T^{k}\), let us count the necessary crossings in \(D\). It is obvious that we have to deal with the possible existence of a subgraph \(T^{l}\in S_{D}\) by which the edges of \(G^{*}\cup T^{i}\cup T^{k}\) can be crossed at most six times.