# On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles
## 2 Cyclic Permutations and Configurations
As in the previous case of four possible configurations, let \(\mathcal{N}_{D}\) denote the set of all existing configurations \(\mathcal{B}_{q}\) in the considered drawing \(D\) such that they are included in the set \(\mathcal{N}=\{\mathcal{B}_{1},\mathcal{B}_{2}\}\). Figure 3: Two possible drawings of \(F^{k}\) with a configuration from \(\mathcal{N}\). Figure 2: Four possible drawings of \(F^{i}\) with a configuration from \(\mathcal{M}\). Now, our aim is to establish a minimum number of edge crossings between two different subgraphs \(F^{i}\) and \(F^{j}\) using the idea of mentioned configurations. For two configurations \(\mathcal{X}\) and \(\mathcal{Y}\) from \(\mathcal{M}_{D}\) (not necessarily different), let \(\operatorname{cr}_{D}(\mathcal{X},\mathcal{Y})\) denote the number of edge crossings in \(D(T^{i}\cup T^{j})\) for two different subgraphs \(T^{i},T^{j}\in R_{D}\) such that \(F^{i}\), \(F^{j}\) have configurations \(\mathcal{X}\), \(\mathcal{Y}\), respectively. We denote by \(\operatorname{cr}(\mathcal{X},\mathcal{Y})\) the minimum value of \(\operatorname{cr}_{D}(\mathcal{X},\mathcal{Y})\) over all pairs \(\mathcal{X}\) and \(\mathcal{Y}\) from \(\mathcal{M}\) among all good drawings \(D\) of the join product \(G^{*}+D_{n}\). In the following, our goal is to determine the lower bounds of \(\operatorname{cr}(\mathcal{X},\mathcal{Y})\) for all possible pairs \(\mathcal{X},\mathcal{Y}\in\mathcal{M}\) and we also partially extend this idea of lower bounds to a subfamily of subgraphs by which the edges \(G^{*}\) are crossed exactly once, that is, for subgraphs with the configurations \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\). At least five interchanges of adjacent elements of \((123654)\) are necessary to obtain cyclic permutation \(\overline{(136542)}=(124563)\) because only one interchange of the adjacent elements of \((123654)\) produces the cyclic permutation \((136542)\). (Let \(T^{x}\) and \(T^{y}\) be any two different subgraphs represented by their \(\operatorname{rot}(t_{x})\) and \(\operatorname{rot}(t_{y})\) of the same length \(m\), where \(m\) is a positive integer of at least \(3\). If the minimum number of interchanges of adjacent elements of \(\operatorname{rot}(t_{x})\) required to produce \(\operatorname{rot}(t_{y})\) is at most \(z\), then \(\operatorname{cr}_{D}(T^{x},T^{y})\geq\left\lfloor\frac{m}{2}\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor-z\), see also [PERSON] [24].) Using this knowledge, the edges of each subgraph \(T^{i}\) with the configuration \(\mathcal{A}_{1}\) of \(F^{i}\) are crossed at least five times by the edges of each subgraph \(T^{j}\) with the configuration \(\mathcal{A}_{3}\) of \(F^{j}\), that is, \(\operatorname{cr}(\mathcal{A}_{1},\mathcal{A}_{3})\geq 5\). The same idea also force \(\operatorname{cr}(\mathcal{A}_{1},\mathcal{A}_{2})\geq 3\), \(\operatorname{cr}(\mathcal{A}_{1},\mathcal{A}_{4})\geq 4\), \(\operatorname{cr}(\mathcal{A}_{2},\mathcal{A}_{3})\geq 4\), \(\operatorname{cr}(\mathcal{A}_{2},\mathcal{A}_{4})\geq 5\), and \(\operatorname{cr}(\mathcal{A}_{3},\mathcal{A}_{4})\geq 3\). Moreover, by a simple discussion, we can verify that the lower bound of \(\operatorname{cr}(\mathcal{A}_{3},\mathcal{A}_{4})\) can be increased up to \(5\). Let \(F^{i}\) be any subgraph with the configuration \(\mathcal{A}_{3}\), and let \(T^{j}\) be some different subgraph from \(R_{D}\) satisfying the restriction \(\operatorname{cr}_{D}(T^{i},T^{j})<5\). If we place the vertex \(t_{j}\) in the region of \(D(G^{*}\cup T^{i})\) with the three vertices \(v_{2}\), \(v_{3}\) and \(v_{4}\) of \(G^{*}\) on its boundary, then the edges \(t_{j}v_{1}\), \(t_{j}v_{5}\) and \(t_{j}v_{6}\) produce exactly one, one and two crossings on the edges of \(T^{i}\), respectively. Thus, \(\operatorname{rot}_{D}(t_{j})=(165432)=\mathcal{A}_{2}\). Other placements of the vertex \(t_{j}\) imply at least five crossings on the edges of \(T^{i}\cup T^{j}\). Clearly, also \(\operatorname{cr}(\mathcal{A}_{p},\mathcal{A}_{p})\geq 6\) for any \(p=1,2,3,4\).