The crossing numbers of join of the special graph on six vertices with path and cycle

Abstract

a b s t r a c tThere are only few results concerning crossing numbers of join of some graphs. In the paper, for the special graph H on six vertices we give the crossing numbers of its join with n isolated vertices as well as with the path P n on n vertices and with the cycle C n .

On the crossing numbers of join products of five graphs of order six with the discrete graph

“…It is also important to note that the crossing numbers of the graphs G + D n are known for few graphs G of order five and six, see e.g. [6,8,[10][11][12][13][14]. In all these cases, the graph G is usually connected and contains at least one cycle.…”

“…In [2,12,13], the properties of cyclic permutations are verified with the help of the software described in [1]. In our opinion, the methods used in [6,8,9] do not suffice for establishing the crossing number of the join product G * + D n . Some parts of proofs can be done with the help of software that generates all cyclic permutations in [1].…”

“…The rotation rot D (t i ) of a vertex t i in the drawing D is the cyclic permutation that records the (cyclic) counterclockwise order in which the edges leave t i , as defined by Hernández-Vélez et al [3]. We use the notation (123456) if the counterclockwise order of the edges incident with the vertex 6 . We have to emphasize that a rotation is a cyclic permutation.…”

“…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 [6].) 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.…”

Joins of 1-planar graphs

“…Recall that the join product (or shortly, join) G + H of two graphs G and H is obtained from vertex-disjoint copies of G and H by adding all edges between V (G) and V (H). In [15], Kulli and Muddebihal characterized planar joins; the crossing numbers of joins of special graphs were studied in [10][11][12][13] and [16] (the crossing number cr(G) of a graph G is the minimum number of crossings in any plane drawing of G).…”