*Article* **Join Products** *K***2,3 +** *Cn*

### **Michal Staš**

Faculty of Electrical Engineering and Informatics, Technical University of Košice, 042 00 Košice, Slovakia; michal.stas@tuke.sk

Received: 7 May 2020; Accepted: 1 June 2020; Published: 5 June 2020

**Abstract:** The crossing number cr(*G*) of a graph *G* is the minimum number of edge crossings over all drawings of *G* in the plane. The main goal of the paper is to state the crossing number of the join product *K*2,3 + *Cn* for the complete bipartite graph *K*2,3, where *Cn* is the cycle on *n* vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph *K*2,3, we also offer the crossing number of the join product of one other graph with the cycle *Cn*.

**Keywords:** graph; join product; crossing number; cyclic permutation; arithmetic mean

### **1. Introduction**

For the first time, P. Turán described the brick factory problem. He was forced to work in a brickyard and his task was to push the bricks of the wagons along the line from the kiln to the storage location. The factory contained several furnaces and storage places, between which sidewalks passed through the floor. Turán found it difficult to move the wagon through the track passage, and in his mind he began to consider how the factory could be redesigned to minimize these crossings. Since then, the topic has steadily grown and research into the number of crosses has become one of the main areas of graph theory. This problem of reducing the number of crossings on the edges of graphs were studied in many areas.

The crossing number cr(*G*) of a simple graph *G* with the vertex set *V*(*G*) and the edge set *E*(*G*) is the minimum possible number of edge crossings in a drawing of *G* in the plane (for the definition of a drawing see [1].) It is easy to see that a drawing with minimum number of crossings (an optimal drawing) is always a good drawing, meaning that no edge crosses itself, no two edges cross more than once, and no two edges incident with the same vertex cross. Let *D* (*D*(*G*)) be a good drawing of the graph *G*. We denote the number of crossings in *D* by cr*D*(*G*). Let *Gi* and *Gj* be edge-disjoint subgraphs of *G*. We denote the number of crossings between edges of *Gi* and edges of *Gj* by cr*D*(*Gi*, *Gj*), and the number of crossings among edges of *Gi* in *D* by cr*D*(*Gi*). It is easy to see that for any three mutually edge-disjoint subgraphs *Gi*, *Gj*, and *Gk* of *G*, the following equations hold:

$$\text{crr}\_D(\mathbf{G}\_i \cup \mathbf{G}\_j) = \text{cr}\_D(\mathbf{G}\_i) + \text{cr}\_D(\mathbf{G}\_j) + \text{cr}\_D(\mathbf{G}\_{i\prime} \mathbf{G}\_j) \,,$$

$$\text{cr}\_D(\mathbf{G}\_i \cup \mathbf{G}\_{j\prime} \mathbf{G}\_k) = \text{cr}\_D(\mathbf{G}\_{i\prime} \mathbf{G}\_k) + \text{cr}\_D(\mathbf{G}\_{j\prime} \mathbf{G}\_k) \,.$$

By Garey and Johnson [2] we already know that calculating the crossing number of a simple graph is an NP-complete problem. Recently, the exact values of the crossing numbers are known only for some special classes of graphs. In [3], Ho gave the characterization for a few multipartite graphs. So, the main purpose of this work is to extend the results concerning this topic for the complete bipartite graph *K*2,3 on five vertices. In this paper we use definitions and notations of the crossing

number theory presented by Klešˇc in [4]. In the proofs we will also use the Kleitman's result [5] on the crossing numbers of the complete bipartite graphs. He estimated that

$$\text{cr}(K\_{m,n}) = \left\lfloor \frac{m}{2} \right\rfloor \left\lfloor \frac{m-1}{2} \right\rfloor \left\lfloor \frac{n}{2} \right\rfloor \left\lfloor \frac{n-1}{2} \right\rfloor, \quad \text{with } \min\{m, n\} \le 6.$$

Again using Kleitman's result [5], the exact values of the crossing numbers for the join product of two paths, the join product of two cycles, and also for the join product of a path and a cycle were proved in [4]. Further, some values for crossing numbers of *G* + *Dn*, *G* + *Pn*, and of *G* + *Cn* for arbitrary graph *G* at most on four vertices are estimated in [6,7]. Let us note that the exact values for the crossing numbers of the join product *G* with *Pn* and *Cn* were also investigated for a few graphs *G* of order five and six in [1,8–12]. In all mentioned cases, the graph *G* contains usually at least one cycle and it is connected.

It is important to note that the methods in this paper will mostly use several combinatorial properties on cyclic permutations. If we place the graph *K*2,3 on the surface of the sphere, from the topological point of view, the resulting number of crossings of *K*2,3 + *Cn* does not matter which of the regions in the subdrawing of *<sup>K</sup>*2,3 <sup>∪</sup> *<sup>T</sup><sup>i</sup>* is unbounded, but on how the subgraph *<sup>T</sup><sup>i</sup>* crosses or does not cross the edges of *K*2,3 (the description of *T<sup>i</sup>* will be justified in Section 2). This representation of *T<sup>i</sup>* can best be described by the idea of a configuration utilizing some cyclic permutation on the pre-numbered vertices of the graph *K*2,3. We introduce a new idea of various form of arithmetic means on a minimum number of crossings between two corresponding subgraphs *T<sup>i</sup>* and *T<sup>j</sup>* . Certain parts of proofs can be also simplified with the help of software which generates all cyclic permutations of five elements due to Berežný and Buša [13].
