Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G* + D-n, where G* is a connected graph isomorphic to K-2,K-2,K-2 \ {e(1),e(2)} obtained by removing two edges e(1),e(2) with a common vertex and a second vertex from the different partitions of the complete tripartite graph K-2,K-2,K-2, and D-n is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs H-k of G* with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles.