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arXiv · 2007.13308

On the sizes of bipartite 1-planar graphs

Yuanqiu Huang·Zhangdong Ouyang·Fengming Dong
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Abstract

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 $n$ ($\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$. Czap, Przybylo and \u{S}krabul\'{a}kov\'{a} proved that if the partite sets of $G$ are of sizes $x$ and $y$, then $m\le 2n+6x-12$ holds for $2\leq x\leq y$, and conjectured that $m\le 2n+4x-12$ holds for $x\ge 3$ and $y\ge 6x-12$. In this paper, we settle their conjecture and our result is even under a weaker condition $2\le x\le y$.

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Yuanqiu Huang, Zhangdong Ouyang, Fengming Dong. 2020-07-27. On the sizes of bipartite 1-planar graphs. https://arxiv.org/abs/2007.13308

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