**2. Possible Drawings of** *K***2,3 and Preliminary Results**

Let us first consider the join product of the complete bipartite graph *K*2,3 with the discrete graph *Dn* considered on *n* vertices. It is not difficult to see that the graph *K*2,3 + *Dn* contains just one copy of the graph *K*2,3 and *n* vertices *t*1, ... , *tn*, where each vertex *ti*, *i* = 1, ... , *n*, is adjacent to every vertex of *<sup>K</sup>*2,3. For 1 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>n</sup>*, let *<sup>T</sup><sup>i</sup>* denote the subgraph which is uniquely induced by the five edges that are incident with the fixed vertex *ti*. This means that the graph *<sup>T</sup>*<sup>1</sup> ∪···∪ *<sup>T</sup><sup>n</sup>* is isomorphic to the graph *K*5,*<sup>n</sup>* and we obtain

$$K\_{2,3} + D\_n = K\_{2,3} \cup K\_{5,n} = K\_{2,3} \cup \left(\bigcup\_{i=1}^n T^i\right). \tag{1}$$

The graph *K*2,3 + *Cn* contains *K*2,3 + *Dn* as a subgraph. For all subgraphs of the graph *K*2,3 + *Cn* which are also subgraphs of the graph *K*2,3 + *nK*<sup>1</sup> we can use the same notations as above. Let *C*<sup>∗</sup> *n* denote the cycle induced on *n* vertices of *K*2,3 + *Cn* but which do not belong to the subgraph *K*2,3. Hence, *C*∗ *<sup>n</sup>* consists of the vertices *t*1, *t*2, ... , *tn* and of the edges {*ti*, *ti*+1} and {*tn*, *t*1} for *i* = 1, ... , *n* − 1. So we get

$$K\_{2,3} + \mathbb{C}\_n = K\_{2,3} \cup K\_{5,n} \cup \mathbb{C}\_n^\* = K\_{2,3} \cup \left(\bigcup\_{i=1}^n T^i\right) \cup \mathbb{C}\_n^\*. \tag{2}$$

In the paper, the definitions and notation of the cyclic permutations and of the corresponding configurations of subgraphs for a good drawing *D* of the graph *K*2,3 + *Dn* presented in [14] are used. By Hernández-Vélez et al. [15], the cyclic permutation that records the (cyclic) counter-clockwise order in which the edges leave a vertex *ti* is said to be the rotation rot*D*(*ti*) of the vertex *ti*. On the basis of this, we use the notation (12345) if the counter-clockwise order the edges incident with the vertex *ti* is *tiv*1, *tiv*2, *tiv*3, *tiv*4, and *tiv*5. Recall that any such rotation is a cyclic permutation. For our research, we will separate all subgraphs *T<sup>i</sup>* of *K*2,3 + *Dn*, *i* = 1, 2, ... , *n*, into three families of subgraphs depending on how many times are edges of *K*2,3 crossed by the edges of the considered subgraph *<sup>T</sup><sup>i</sup>* in *<sup>D</sup>*. For *<sup>i</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*, let *RD* <sup>=</sup> {*T<sup>i</sup>* : cr*D*(*K*2,3, *<sup>T</sup><sup>i</sup>* ) = <sup>0</sup>}, and *SD* <sup>=</sup> {*T<sup>i</sup>* : cr*D*(*K*2,3, *<sup>T</sup><sup>i</sup>* ) = 1}. The edges of *<sup>K</sup>*2,3 are crossed at least twice by each other subgraph *<sup>T</sup><sup>i</sup>* in *<sup>D</sup>*. For *<sup>T</sup><sup>i</sup>* <sup>∈</sup> *RD* <sup>∪</sup> *SD*, let

*<sup>F</sup><sup>i</sup>* denote the subgraph *<sup>K</sup>*2,3 <sup>∪</sup> *<sup>T</sup><sup>i</sup>* , *i* ∈ {1, 2, ... , *n*}, of *K*2,3 + *Dn*. Clearly, the idea of dividing the subgraphs *T<sup>i</sup>* into three mentioned families is also retained in all drawings of the graph *K*2,3 + *Cn*. In [14], there are two possible non isomorphic drawings of the graph *K*2,3, but only with the possibility of obtaining a subgraph *<sup>T</sup><sup>i</sup>* <sup>∈</sup> *RD* in *<sup>D</sup>*. Due to the arguments in the proof of Theorem 2, if we wanted to get an optimal drawing *D* of *K*2,3 + *Cn*, then the subdrawing *D*(*K*2,3) of the graph *K*2,3 induced by *D* with at least three crossings among the edges of *K*2,3 forces that the set *RD* must be nonempty. But, in the cases of cr*D*(*K*2,3) ≤ 2, just one of the sets *RD* or *SD* can be empty. With these assumptions, we obtain four non isomorphic drawings of the graph *K*2,3 as shown in Figure 1. The vertex notation of *K*2,3 will be substantiated later in all mentioned drawings, and wherein two disjoint independent sets of vertices of the complete bipartite graph *K*2,3 will be also highlighted by filled and non filled rings.

**Figure 1.** Four possible non isomorphic drawings of the graph *K*2,3. (**a**) the planar drawing of *K*2,3; (**b**) the drawing of *K*2,3 with one crossing on edges of *K*2,3; (**c**) the drawing of *K*2,3 with cr*D*(*K*2,3) = 2; (**d**) the drawing of *K*2,3 with cr*D*(*K*2,3) = 3.

### **3. The Crossing Number of** *K***2,3 +** *Cn*

In the proofs of the paper, several parts are based on the Theorem 1 presented in [14].

$$\text{Theorem 1. } \text{cr}(K\_{2,3} + D\_n) = 4 \left\lfloor \frac{n}{2} \right\rfloor \left\lfloor \frac{n-1}{2} \right\rfloor + n \text{ for any } n \ge 1.$$

Now we are able to prove the main results of the paper. The exact values of the crossing numbers of the small graphs *K*2,3 + *C*3, *K*2,3 + *C*4, and *K*2,3 + *C*<sup>5</sup> can be estimated using the algorithm located on the website http://crossings.uos.de/ provided that it uses an ILP formulation based on Kuratowski subgraphs and also generates verifiable formal proofs, for more see [16].
