Cyclic permutations in determining crossing numbers

Abstract

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. Recently, the crossing numbers of join products of two graphs have been studied. In the paper, we extend know results concerning crossing numbers of join products of small graphs with discrete graphs. The crossing number of the join product G * + D n for the disconnected graph G * consisting of five vertices and of three edges incident with the same vertex is given. Up to now, the crossing numbers of G + D n were done only for connected graphs G. In the paper also the crossing numbers of G * + P n and G * + C n are given. The paper concludes by giving the crossing numbers of the graphs H + D n , H + P n , and H + C n for four different graphs H with |E(H)| ≤ |V (H)|. The methods used in the paper are new. They are based on combinatorial properties of cyclic permutations.

The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

“…This contradicts the fact that there are, at most, n 2 crossings on the edge v 1 v 5 using the already well-known result from the previous case by [11]. Finally, if all vertices t i are placed in the region of D(G 11 ) with four vertices v 1 , v 2 , v 4 , and v 5 of G 11 on its boundary, then only one of the edges v 1 v 5 and v 2 v 5 can be crossed by any subgraph T i ∈ S D again by Corollary 1.…”

“…Let us turn to the possibility of an existence of vertex t j of the cycle C * n in some region of D(G 11 ) with three vertices of G 11 on its boundary, that is, two different edges of C * n cross one of the edges v 1 v 5 or v 3 v 5 in D again by Corollary 2. Since there are two additional crossings on one of these two edges of the graph G 11 , the mentioned result [11] enforces r ≥ n 2 + 1 ≥ 4 for n at least 5. Let D be the subdrawing of G 11 + D n induced by D without the edges of C * n .…”

“…Since the set R D ∪ S D is nonempty and no edge of the path P * n is crossed in the drawing D, all vertices t i of P * n are placed in the region of subdrawing D(G 11 ) with five vertices v 1 , v 2 , v 3 , v 4 , and v 5 of G 11 on its boundary. By Klešč and Staš [11], it was proved that cr(G 0…”

“…Finally, if all vertices t i are placed in the region of D(G 11 ) with four vertices v 1 , v 2 , v 4 , and v 5 of G 11 on its boundary, then only one of the edges v 1 v 5 and v 2 v 5 can be crossed by any subgraph T i ∈ S D again by Corollary 1. The authors in [10,11] proved that cr(G 5…”

“…Notice that a lot of the exact values for crossing numbers of G + D n , G + P n , and of G + C n for arbitrary graph G at most on four vertices were estimated in [5,6]. The crossing numbers of the join product of many graphs G on five and six vertices with P n and C n were also investigated in [1,[7][8][9][10][11][12][13][14][15].…”