The maximum number of cliques in a graph embedded in a surface

TL;DR

This paper determines the maximum number of cliques in graphs embedded on surfaces, characterizing extremal graphs and providing bounds that depend on the surface's properties.

Contribution

It characterizes extremal graphs and establishes bounds for the maximum number of cliques in surface-embedded graphs, with exact results for specific surfaces.

Findings

Bounds between $8(n- ext{omega})+2^{ ext{omega}}$ and $8n+\frac{3}{2} 2^{\text{omega}}+o(2^{\text{omega}})$

Exact maximum clique counts for certain surfaces

Characterization of extremal graphs for the problem

Abstract

This paper studies the following question: Given a surface $\Sigma$ and an integer $n$, what is the maximum number of cliques in an $n$-vertex graph embeddable in $\Sigma$? We characterise the extremal graphs for this question, and prove that the answer is between $8(n-\omega)+2^{\omega}$ and $8n+{3/2} 2^{\omega}+o(2^{\omega})$, where $\omega$ is the maximum integer such that the complete graph $K_\omega$ embeds in $\Sigma$. For the surfaces $\mathbb{S}_0$, $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{N}_3$ and $\mathbb{N}_4$ we establish an exact answer.