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On the sizes of bipartite 1-planar graphs
A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let G be a bipartite 1-planar graph with n (n ≥4) vertices and m edges. Karpov showed that m ≤ 3n - 8 holds for even n ≥ 8 and m ≤3n - 9 holds for odd n ≥ 7. Czap, Przybylo and Škrabul’áková proved that if the partite sets of G are of sizes x and y, then m ≤ 2n+6x-12 holds for 2 ≤ x ≤ y, and conjectured that m ≤ 2n + 4x - 12 holds for x ≥ 3 and y ≥ 6x - 12. In this paper, we settle their conjecture and our result is even under a weaker condition 2 ≤ x ≤ y.