ON THE CROSSING NUMBER OF THE JOIN OF FIVE VERTEX GRAPH WITH THE DISCRETE GRAPH Dn

Abstract

In this paper, we show the values of crossing numbers for join products of graph G on five vertices with the discrete graph D n and the path P n on n vertices. The proof is done with the help of software. The software generates all cyclic permutations for a given number n. For cyclic permutations, P 1 -P m will create a graph in which to calculate the distances between all vertices of the graph. These distances are used in proof of crossing numbers of presented graphs.

Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

“…We remark that the crossing numbers of the graphs H 1 + D n and H 3 + D n were already obtained by Berežný and Staš [4], and Klešč and Schrötter [7], respectively. Moreover, into the drawing in Figure 4b, it is possible to add n edges, which form the path P n , n ≥ 2 on the vertices of D n without another crossing.…”

“…Moreover, the exact values for crossing numbers of G + D n and G + P n for all graphs G of order at most four are given in [3]. Furthermore, the crossing numbers of the graphs G + D n are known for a few graphs G of order five and six in [4][5][6][7][8][9][10]. In all of these cases, the graph G is connected and contains at least one cycle.…”

“…Similar methods were partially used for the first time in the papers [8,13]. In [4,9,10,14], the properties of cyclic permutations were also verified with the help of software in [15]. In our opinion, the methods used in [3,5,7] do not allow establishing the crossing number of the join product G + D n .…”

Determining Crossing Numbers of Graphs of Order Six Using Cyclic Permutations

“…The purpose of this article is to extend the known results in [2,[7][8][9][10][11] and [12] by adding another graph. The methods are based on combinatorial properties of cyclic permutations.…”

On the Crossing Number of the Join of the Wheel on Five Vertices With the Discrete Graph

“…Somewhat similar ideas were used in [8,17]. In [2,3,18,19], the properties of cyclic permutations are verified with the help of the software in [1]. In our opinion, the methods used in [11,14,15] do not suffice for establishing the crossing number of the join product W 4 + D n .…”