The maximum number of $k$-cliques of 7-connected 1-planar graphs

Abstract

In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the high-connectivity end, we prove that every $n$-vertex 7-connected 1-planar graph has at most $4n-12$ edges, $4n-16$ triangles, and $n-6$ copies of $K_4$. Hence the total number of cliques is at most $10n-33$. All bounds are sharp for infinitely many values of $n$.