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

Abstract

The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph D n and with the path P n on n vertices for three other graphs.

On the crossing number of join product of the discrete graph with special graphs of order five

“…In all these cases, the graph G is connected and contains at least one cycle. The crossing numbers of the join product G + D n are known only for some disconnected graphs G, and so the purpose of this article is to extend the known results concerning this topic to new disconnected graphs, see [2] and [15].…”

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.…”

On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

“…We present a new technique of recalculating the number of crossings due to the combined fixation of different types of subgraphs in an effort to achieve the crossings numbers of G + P n and G + C n also for all graphs G of orders five and six. Of course, cr(G + P n ) and cr(G + C n ) are already known for a lot of connected graphs G of orders five and six [1,[9][10][11][12][13][14][15][16][17], but only for some disconnected graphs [18][19][20].…”