Counting cliques in $1$-planar graphs

TL;DR

This paper determines the maximum number of cliques of all sizes in 1-planar graphs and characterizes the extremal graphs, advancing understanding of clique counts in this graph class.

Contribution

It provides exact maximum clique counts and characterizes extremal graphs in the class of 1-planar graphs, a previously less understood graph class.

Findings

Maximum total number of cliques in 1-planar graphs is determined.

Maximum number of cliques of any fixed size is established.

Extremal graphs achieving these maxima are characterized.

Abstract

The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.