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

TL;DR

This paper establishes sharp upper bounds on the number of cliques in 7-connected 1-planar graphs, extending extremal graph theory results to highly connected cases.

Contribution

It proves the maximum number of edges, triangles, and $K_4$ subgraphs in 7-connected 1-planar graphs, with bounds that are tight for infinitely many values of n.

Findings

Maximum edges in 7-connected 1-planar graphs is 4n-12.

Maximum triangles in such graphs is 4n-16.

Total cliques are at most 10n-33.

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 $n$.