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.
Highlights
The crossing number cr(G) of a simple graph G with the vertex set V(G) and the edge set E(G) is the minimum possible number of edge crossings in a drawing of G in the plane. (For the definition of a drawing, see Klešc [1].) One can verify that a drawing with the minimum number of crossings is always a good drawing, meaning that no two edges cross more than once, no edge crosses itself, and no two edges incident with the same vertex cross
We present a new technique regarding the use of knowledge from the subgraphs whose values of crossing numbers are already known
The proofs are done with the help of a lot of well-known exact values for the crossing numbers of the join products of five subgraphs of G11 with paths and cycles
Summary
The crossing number cr(G) of a simple graph G with the vertex set V(G) and the edge set E(G) is the minimum possible number of edge crossings in a drawing of G in the plane. (For the definition of a drawing, see Klešc [1].) One can verify that a drawing with the minimum number of crossings (an optimal drawing) is always a good drawing, meaning that no two edges cross more than once, no edge crosses itself, and no two edges incident with the same vertex cross. The results of G6 + Pn, G9 + Pn, G11 + Pn, G14 + Pn and of G6 + Cn, G9 + Cn, G11 + Cn, G14 + Cn can be used to determine the crossing number of the join product of the most complicated graph K5 with the path and the cycle on n vertices For this purpose, we present a new technique regarding the use of knowledge from the subgraphs whose values of crossing numbers are already known. The proofs are done with the help of a lot of well-known exact values for the crossing numbers of the join products of five subgraphs of G11 with paths and cycles In certain parts of the presented proofs, it is possible to simplify the procedure with the help of software generating all cyclic permutations of five elements and its description can be found in Berežný and Buša [20]
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