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The crossing numbers of join products of four graphs of order five with paths and cycles

Staš Michal, Timková Mária

Opuscula Mathematica

2023

Vol. 43, no. 6

865--883

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products G∗ + Pn and G∗ + Cn for the disconnected graph G∗ consisting of the complete tripartite graph K1,1,2 and one isolated vertex are given, where Pn and Cn are the path and the cycle on n vertices, respectively. In the paper also the crossing numbers of H∗ + Pn and H∗ + Cn are determined, where H∗ is isomorphic to the complete tripartite graph K1,1,3. Finally, by adding new edges to the graphs G∗ and H∗, we are able to obtain crossing numbers of join products of two other graphs G1 and H1 with paths and cycles.