Books versus Triangles near the n/6 Threshold

TL;DR

This paper proves a conjecture about the minimum number of triangles in large graphs with a given book number, confirming the structure of extremal graphs near the lower bound of the book number range.

Contribution

It establishes the conjecture for all book numbers between n/6 and (1/6 + epsilon)n, confirming the structure of extremal graphs in this range.

Findings

Confirmed the conjecture for all b in [n/6, (1/6 + epsilon)n]

Proved extremal graphs are close to a blow-up of the 3-prism

Established a stability theorem for the structure of extremal graphs

Abstract

The book number $b(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. A strengthening of Mantel's theorem due to Rademacher states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor$ edges contains at least $\lfloor n/2\rfloor$ triangles. Another strengthening, initiated by Erd\H{o}s, asserts that every such graph $G$ satisfies $b(G)\ge n/6$. Motivated by these results, Mubayi studied the tradeoff between the total number of triangles and the book number in such graphs, and asymptotically resolved the problem when $n/4\le b(G)\le n/2$. Conlon, Fox, and Sudakov conjectured that, for $n/6\le b< n/4$, every $n$-vertex graph with at least $\lfloor n^2/4\rfloor$ edges and book number at most $b$, other than the balanced complete bipartite graph, has at least $b^2(n-4b)$ triangles, with equality only for the blow-up $S_{b,n}$ of the $3$-prism. They proved the conjecture when $b$ lies in an interval with endpoint $n/4$, and also at the endpoint $b=n/6$, where they asked whether it remains valid in an interval containing this endpoint. In this paper, we answer this question affirmatively. We show that there exists a constant $\varepsilon>0$ such that the conjecture holds for all $n/6\le b\le (1/6+\varepsilon)n$. Our proof first establishes a stability theorem showing that every extremal graph is close to a blow-up of the $3$-prism, and then uses a detailed parameter analysis to force the exact six-partite structure.