Extremal 1-planar graphs without k-cliques

TL;DR

This paper improves bounds on the maximum number of edges in k-clique-free 1-planar graphs for k=3,4,5, establishing tight bounds and extending previous results in planar Turán-type problems.

Contribution

It strengthens existing bounds on edges in K3, K4, and K5-free 1-planar graphs, providing tight bounds for all large enough n.

Findings

Bound for K3-free 1-planar graphs improved to 3n - 8 edges.

Bound for K4-free 1-planar graphs established as floor(7n/2) - 7 edges.

Bound for K5-free 1-planar graphs set at 4n - 8 edges.

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 $n \ge 10$.