
TL;DR
This paper explores the relationship between the size of the largest book in a graph and the minimum number of triangles it contains, establishing new asymptotic bounds and using advanced combinatorial theorems.
Contribution
It provides a new asymptotic bound linking book size constraints to triangle counts, combining and extending classical results in extremal graph theory.
Findings
For th, the number of triangles is at least th(1-th) n^2/4 - o(n^2).
Existence of eta > 0 such that for th, the graph contains at least eta n^3 triangles.
The results are asymptotically sharp and use the Ruzsa-Szemeredi theorem.
Abstract
A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and \lfloor n^2/4\rfloor +1 edges. Rademacher proved that G contains at least \lfloor n/2\rfloor triangles, and Erdos conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following "linear combination" of these two results. Suppose that \alpha\in (1/2, 1) and the maximum size of a book in G is less than \alpha n/2. Then G contains at least \alpha(1-\alpha) n^2/4 - o(n^2) triangles as n approaches infinity. This is asymptotically sharp. On the other hand, for every \alpha\in (1/3, 1/2), there exists \beta>0 such that G contains at least \beta n^3 triangles. It remains an open problem to determine the largest possible \beta in terms of \alpha. Our proof uses the Ruzsa-Szemeredi theorem.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
