# 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  
Let us suppose a graph \(G\) on six vertices with the vertex set \(V(G)=\{v_{1},v_{2},\ldots,v_{6}\}\), and let \(t_{1},t_{2},\ldots,t_{n},t_{1}\) be the vertex notation of the \(n\)-cycle \(C_{n}\) for \(n\geq 3\). The join product \(G+C_{n}\) consists of one copy of the graph \(G\), one copy of the cycle \(C_{n}\), and the edges joining each vertex of \(G\) with each vertex of \(C_{n}\). Let \(C_{n}^{*}\) denote the cycle as a subgraph of \(G+C_{n}\) induced on the vertices of \(C_{n}\) not belonging to the subgraph \(G\). The subdrawing \(D(C_{n}^{*})\) induced by any good drawing \(D\) of \(G+C_{n}\) represents some drawing of \(C_{n}\). For the vertices \(v_{1},v_{2},\ldots,v_{6}\) of graph \(G\), we denote by \(T^{v_{1}}\) the subgraph induced by \(n\) edges joining the vertex \(v_{i}\) with \(n\) vertices of \(C_{n}^{*}\). The edges joining six vertices of \(G\) with \(n\) vertices of \(C_{n}\) form the complete bipartite graph \(K_{6,n}\), and so  
\[G+C_{n}=G\cup K_{6,n}\cup C_{n}^{*}=G\cup\left(\bigcup_{i=1}^{6}T^{v_{i}} \right)\cup C_{n}^{*}. \tag{8}\]  
In the proofs of the theorems, the following two statements regarding some restricted subdrawings of \(G+C_{n}\) are useful. Lemma 5 (See [6] Lemma 2.2): _Let \(D\) be a good drawing of \(D_{m}+C_{n}\), \(m\geq 2\), \(n\geq 3\) in which no edge of \(C_{n}^{*}\) is crossed, and \(C_{n}^{*}\) does not separate the other vertices of the graph. Then, for all \(i,j=1,2,\ldots,m\), two different subgraphs \(T^{v_{i}}\) and \(T^{v_{j}}\) cross each other in \(D\) at least \(\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\) times._  
Lemma 6 (See [28] Lemma 1): _Let \(G\) be a graph of order \(m\), \(m\geq 1\). In an optimal drawing of the join product \(G+C_{n}\), \(n\geq 3\), the edges of \(C_{n}^{*}\) do not cross each other._  
Theorem 6: \(\operatorname{cr}(G^{*}+C_{n})=6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor \frac{n-1}{2}\right\rfloor+2\left\lfloor\frac{n}{2}\right\rfloor+2\) _for \(n\geq 3\)._  
Proof: In both Figures 6 and 7, it is possible to add the edge \(t_{1}t_{n}\) that creates the cycle \(C_{n}^{*}\) on the vertices of \(P_{n}^{*}\) with just two additional crossings, i.e., \(C_{n}^{*}\) is crossed by two edges \(v_{3}v_{4}\) and \(v_{3}v_{6}\) of the graph \(G^{*}\). So, \(\operatorname{cr}(G^{*}+C_{n})\leq 6\left\lfloor\frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor+2\left\lfloor\frac{n}{2}\right\rfloor+2\). To prove the lower border, let \(D\) be a good drawing of \(G^{*}+C_{n}\) 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. By Theorem 1, at most, one edge of \(C_{n}^{*}\) can be crossed in \(D\), and we can also suppose that the edges of \(C_{n}^{*}\) do not cross each other using Lemma 6. The induced subdrawing \(D(C_{n}^{*})\) divides the plane into two regions with at least four vertices of the 4-cycle of \(G^{*}\) in one of them, and so three possible cases may occur:  
**Case 1:** No edge of \(C_{n}^{*}\) crosses any edge of \(G^{*}\), that is, all six vertices of \(G^{*}\) must be placed in one region of the subdrawing \(D(C_{n}^{*})\). As at least five different subgraphs \(T^{v_{i}}\) cannot cross the edges of \(C_{n}^{*}\), any two such different subgraphs \(T^{v_{i}}\) and \(T^{v_{j}}\) cross each other at least \(\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\) times by Lemma 5. This forces at least \(\left(\frac{5}{2}\right)\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor \frac{n-1}{2}\right\rfloor>6\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor \frac{n-1}{2}\right\rfloor+2\left\lfloor\frac{n}{2}\right\rfloor+1\) crossings in \(D\). **Case 2:** Some edge of \(C_{n}^{*}\) crosses the edge \(v_{1}v_{2}\) of \(G^{*}\).