The crossing number of G ▭ C n for the graph G on six vertices

Abstract

ABSTRACT. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G C n for some graphs G on five and six vertices and the cycle C n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G C n , where G is a specific graph on six vertices.

The crossing numbers of join products of eight graphs of order six with paths and cycles

“…, it is not difficult to verify that the conditions (2), ( 3) and ( 4), (5) are fulfilled for the drawing of H 1 in Figure 8 the number of crossings in (7) confirms a contradiction with the assumption in D for all n at least 4. For n = 3, we obtain also the contradiction with the number of crossings in D except for the case of the drawing of H 1 in Figure 8 (a) with T 1 , T 2 ∈ R 0 and T 3 by which the edges of H 1 are crossed exactly once, but the same discussion as in Case 1 forces at least 11 crossings in D again.…”

“…The crossing numbers of the join products of the paths and the cycles with all graphs of order at most four have been well-known for a long time by M. Klešč [10,11], and M. Klešč and Š. Schr ötter [14], and therefore it is understandable that our immediate goal is to establish the exact values for the crossing numbers of G + P n and G + C n also for all graphs G of order five and six. Of course, the crossing numbers of G + P n and G + C n are already known for a lot of graphs G of order five and six (see [3,5,6,9,12,15,[17][18][19][20]24]). In all these cases, the graph G is connected and contains usually at least one cycle.…”

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

“…In the ensuing years, significant effort has gone into extending these results to include graphs on more vertices; in particular five and six vertices. The pioneering work in this area was by Klešč and his various co-authors [6][7][8][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] who have spent the last three decades handling these cases, often on a graph-by-graph basis, requiring ad-hoc proofs that exploit the specific graph structure of the graphs in question. In the last fifteen years, a large number of other researchers have also contributed to this field.…”

“…6 46 P n . Of course, this kind of approach is only useful when the upper bound coincides with an established lower bound for a subgraph.…”

The crossing numbers of products with cycles

“…In [7],the values of crossing numbers for sevetal Cartesian products of cycles and six-edge graphs G on six vertices are presented. In [7] and [5] are given the crossing numbers for Cartesian products of cycles and two seven-edge graphs G on six vertices. In this paper, we give the crossing number of the Cartesian products G 2C n for two graphs G on six vertices and fixed number n.…”