Abstract
The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .
Highlights
If we place the graph K2,3 on the surface of the sphere, from the topological point of view, the resulting number of crossings of K2,3 + Cn does not matter which of the regions in the subdrawing of K2,3 ∪ T i is unbounded, but on how the subgraph T i crosses or does not cross the edges of K2,3. This representation of T i can best be described by the idea of a configuration utilizing some cyclic permutation on the pre-numbered vertices of the graph K2,3
We introduce a new idea of various form of arithmetic means on a minimum number of crossings between two corresponding subgraphs T i and T j
If there is a T i ∈ R D, the edges of K2,3 ∪ T i are crossed at least four times by any subgraph T j with the placement of the vertex t j in the outer region of D (K2,3 ), which yields that the similar idea as in the previous subcase can be used by fixing the subgraph K2,3 ∪ T i crD (K2,3 + Cn ) ≥ 4
Summary
This representation of T i can best be described by the idea of a configuration utilizing some cyclic permutation on the pre-numbered vertices of the graph K2,3. For 1 ≤ i ≤ n, let T i denote the subgraph which is uniquely induced by the five edges that are incident with the fixed vertex ti This means that the graph T 1 ∪ · · · ∪ T n is isomorphic to the graph.
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