Counting copies of a fixed subgraph in $F$-free graphs

TL;DR

This paper advances the understanding of the maximum number of subgraphs in $F$-free graphs, providing new bounds, exact asymptotics for specific cases, and characterizing graphs that influence the growth rate of this generalized Turán function.

Contribution

It introduces new bounds for the generalized Turán function, determines asymptotic values for specific subgraphs, and characterizes graphs affecting the linearity of the function.

Findings

Established new bounds for $ex(n,H,F)$.

Determined asymptotics for $ex(n,P_k,K_{2,t})$ and $ex(n,C_k,K_{2,t})$.

Characterized graphs $F$ that make $ex(n,C_k,F)$ linear in $n$.

Abstract

Fix graphs $F$ and $H$ and let $ex(n,H,F)$ denote the maximum possible number of copies of the graph $H$ in an $n$-vertex $F$-free graph. The systematic study of this function was initiated by Alon and Shikhelman [{\it J. Comb. Theory, B}. {\bf 121} (2016)]. In this paper, we give new general bounds concerning this generalized Tur\'an function. We also determine $ex(n,P_k,K_{2,t})$ (where $P_k$ is a path on $k$ vertices) and $ex(n,C_k,K_{2,t})$ asymptotically for every $k$ and $t$. For example, it is shown that for $t \geq 2$ and $k\geq 5$ we have $ex(n,C_k,K_{2,t})=\left(\frac{1}{2k}+o(1)\right)(t-1)^{k/2}n^{k/2}$. We also characterize the graphs $F$ that cause the function $ex(n,C_k,F)$ to be linear in $n$. In the final section we discuss a connection between the function $ex(n,H,F)$ and Berge hypergraph problems.