The crossing numbers of certain Cartesian products
In this article we determine the crossing numbers of the Cartesian products of given three graphs on five vertices with paths.
The crossing numbers of products of a 5-vertex graph with paths and cycles
There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with "small" graphs. Let H be the 5-vertex graph defined from K₅ by removing three edges incident with a common vertex. In this paper, we extend the earlier results to the Cartesian products of H × Pₙ and H × Cₙ, showing that in the general case the corresponding crossing numbers are 3n-1, and 3n for even n or 3n+1 if n is odd.
Some crossing numbers of products of cycles
The exact values of crossing numbers of the Cartesian products of four special graphs of order five with cycles are given and, in addition, all known crossing numbers of Cartesian products of cycles with connected graphs on five vertices are summarized.
On the crossing numbers of Cartesian products of stars and paths or cycles
The Crossing Numbers of Products of Path with Graphs of Order Six
The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G⃞Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G⃞Pn for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian...
On the Crossing Numbers of Cartesian Products of Stars and Graphs of Order Six
The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. According to their special structure, the class of Cartesian products of two graphs is one of few graph classes for which some exact values of crossing numbers were obtained. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. Moreover, except of six graphs, the crossing numbers of Cartesian products G⃞K1,n for all other...
On the crossing numbers of G □ Cₙ for graphs G on six vertices
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☐Cₙ for some graphs G on five and six vertices and the cycle Cₙ are also given. In this paper, we extend these results by determining crossing numbers of Cartesian products G☐Cₙ for some connected graphs G of order six with six and seven edges. In addition, we collect known results concerning crossing numbers of G☐Cₙ for graphs G on six vertices.
The crossing numbers of join products of paths with graphs of order four
Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers...
On the Crossing Numbers of Cartesian Products of Wheels and Trees
Bokal developed an innovative method for finding the crossing numbers of Cartesian product of two arbitrarily large graphs. In this article, the crossing number of the join product of stars and cycles are given. Afterwards, using Bokal’s zip product operation, the crossing numbers of the Cartesian products of the wheel Wn and all trees T with maximum degree at most five are established.
Page 1