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arXiv · 2604.21589

Extremal 1-planar graphs without k-cliques

Licheng Zhang·Yuanqiu Huang·Fengming Dong
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Abstract

In 2016, Dowden initiated the study of planar Tur\'an-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In this paper, we strengthen this bound to $3n - 8$, which is tight for all even $n \ge 8$. Furthermore, we show that every $K_4$-free $1$-planar graph on $n \ge 3$ vertices has at most $\bigl\lfloor \tfrac{7n}{2} \bigr\rfloor - 7$ edges, and this bound is tight for all integers $n \ge 9$. We also prove that every $K_5$-free $1$-planar graph on $n \ge 3$ vertices has at most $4n - 8$ edges, which is tight for $n = 8$ and for all integers $n \ge 10$.

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Licheng Zhang, Yuanqiu Huang, Fengming Dong. 2026-04-23. Extremal 1-planar graphs without k-cliques. https://arxiv.org/abs/2604.21589

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