Extremal 1-planar graphs without k-cliques
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915954307366912 |
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| author | Zhang, Licheng Huang, Yuanqiu Dong, Fengming |
| author_facet | Zhang, Licheng Huang, Yuanqiu Dong, Fengming |
| contents | In 2016, Dowden initiated the study of planar Turán-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 $n \ge 10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal 1-planar graphs without k-cliques Zhang, Licheng Huang, Yuanqiu Dong, Fengming Combinatorics In 2016, Dowden initiated the study of planar Turán-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 $n \ge 10$. |
| title | Extremal 1-planar graphs without k-cliques |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.21589 |