On the crossing numbers of join products of W4 + Pn and W4 + Cn

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 aim of the paper is to give the crossing number of the join product W4 + Pn and W4 + Cn for the wheel W4 on five vertices, where Pn and Cn are the path and the cycle on n vertices, respectively. Yue et al. conjectured that the crossing number of Wm + Cn is equal to [formula], for all m,n ≥ 3, and where the Zarankiewicz’s number[formula] is defined for n ≥ 1. Recently, this conjecture was proved for W3 + Cn by Klesc. We establish the validity of this conjecture for W4 + Cn and we also offer a new conjecture for the crossing number of the join product Wm + Pn for m ≥ 3 and n ≥ 2.

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Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).