Maximizing the density of $K_t$'s in graphs of bounded degree and clique number

TL;DR

This paper investigates the maximum density of complete subgraphs in graphs with bounded degree and clique number, extending classical results and identifying extremal structures for various parameter ranges.

Contribution

It combines degree and clique constraints to determine the maximum number of $K_t$ copies per vertex, providing exact results for specific parameter families and showing the non-existence of universal extremal graphs.

Findings

Asymptotic formula for maximum $K_t$ copies per vertex under combined constraints.

Exact determination of extremal functions for certain $( ext{degree}, ext{clique})$ pairs.

Existence of pairs where no single extremal graph maximizes all $K_t$ counts.

Abstract

Zykov showed in 1949 that among graphs on $n$ vertices with clique number $\omega(G) \le \omega$, the Tur\'an graph $T_{\omega}(n)$ maximizes not only the number of edges but also the number of copies of $K_t$ for each size $t$. The problem of maximizing the number of copies of $K_t$ has also been studied within other classes of graphs, such as those on $n$ vertices with maximum degree $\Delta(G) \le \Delta$. We combine these restrictions and investigate which graphs with $\Delta(G) \le \Delta$ and $\omega(G) \le \omega$ maximize the number of copies of $K_t$ per vertex. We define $f_t(\Delta,\omega)$ as the supremum of $\rho_t$, the number of copies of $K_t$ per vertex, among such graphs, and show for fixed $t$ and $\omega$ that $f_t(\Delta,\omega) = (1+o(1))\rho_t(T_{\omega}(\Delta+\lfloor\frac{\Delta}{\omega-1}\rfloor))$. For two infinite families of pairs $(\Delta,\omega)$, we determine $f_t(\Delta,\omega)$ exactly for all $t\ge 3$. For another we determine $f_t(\Delta,\omega)$ exactly for the two largest possible clique sizes. Finally, we demonstrate that not every pair $(\Delta,\omega)$ has an extremal graph that simultaneously maximizes the number of copies of $K_t$ per vertex for every size $t$.