Counting cliques without generalized theta graphs

TL;DR

This paper characterizes the maximum number of triangles in graphs avoiding generalized theta graphs, extending Turán-type results and connecting to the triangle removal lemma.

Contribution

It provides a complete classification of the Turán number for generalized theta graphs and determines exact values for certain edge-critical cases.

Findings

Classifies when the Turán number is quadratic, nearly quadratic, or subquadratic.

Provides exact Turán numbers for graphs avoiding edge-critical generalized theta graphs.

Extends recent results on generalized Turán problems.

Abstract

The \textit{generalized Tur\'an number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemer\'edi triangle removal lemma. Motivated by this, in this paper, we study the generalized Tur\'an problem for generalized theta graphs, a natural extension of book graphs. Our main result provides a complete characterization of the magnitude of $\ex(n,K_3,H)$ when $H$ is a generalized theta graph, indicating when it is quadratic, when it is nearly quadratic, and when it is subquadratic. Furthermore, as an application, we obtain the exact value of $\ex(n, K_r, kF)$, where $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results.