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

Abstract

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

"On the crossing number of the join of the wheel on six vertices with a path"

“…Recently, this conjecture was proved for the crossing numbers of join products W 3 + P n and W 4 + P n by Klešč and Schr ötter [17] and by Staš and Valiska [25], respectively. Results by Klešč [11] and [12] establish the conjecture for W m + P 2 and W m + P 3 , and by Staš [23] for W m + P 4 .…”

“…Further, the graph W m + P 4 is isomorphic to the join product of the cycle C m with the graph K 1,4 + 3e obtained by adding three non incident edges with the same vertex to the complete bipartite graph K 1,4 . Using the result of Staš [23], the crossing numbers of the graphs (K 1,4 + 3e) + C m are given by 4 m 2 m−1 2…”

“…Using Kleitman's result, the crossing numbers for the join product of paths with all graphs of order four were studied by Klešč [11] and Klešč and Schr ötter [17]. The exact values for the crossing numbers of G + P n for some graphs G on five vertices are determined in [16,18,22,23,24,25]. The crossing numbers of the join product G + P n are known only for a few graphs G of order six, and so the purpose of this article is to extend the known results concerning this topic to new connected graphs, see [2,5,6,10,13,20].…”

Disconnected spanning subgraphs of paths in the join products with cycles

“…The crossing numbers of G + C n are already known for a lot of graphs G of order five and six [4,6,9,10,8,12,13,14,17]. In all these cases, the graph G is connected and contains usually 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].…”