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

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 $t$.