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Most probably trangle-free graphs

Yuhang Bai, Gyula O. H. Katona, Zixuan Yang

Abstract

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor n^2/4\right\rfloor$ edges--a problem known as the Erdős-Rademacher problem. In this paper, we propose a probabilistic variant of this classic problem. Specifically, given an $n$-vertex graph $G$ with $\left\lfloor n^2/4\right\rfloor+i$ ($i>0$) edges, we choose the edges of $G$ independently with probability $p$, and the resulting new graph is triangle-free with a certain probability. Our goal is to maximize this probability by choosing $G$ appropriately. For the case where $G$ has $ \left\lfloor n^2/4\right\rfloor +1$ edges, we determine the exact maximum probability.

Most probably trangle-free graphs

Abstract

The celebrated Mantel's theorem states that any triangle-free graph on vertices contains at most edges. It is natural to ask how many triangles must exist in a graph with more than edges--a problem known as the Erdős-Rademacher problem. In this paper, we propose a probabilistic variant of this classic problem. Specifically, given an -vertex graph with () edges, we choose the edges of independently with probability , and the resulting new graph is triangle-free with a certain probability. Our goal is to maximize this probability by choosing appropriately. For the case where has edges, we determine the exact maximum probability.
Paper Structure (5 sections, 6 theorems, 33 equations)

This paper contains 5 sections, 6 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:triangle']}
  4. Conclusions
  5. Funding statement

Key Result

Theorem 1

Let $G$ be an $n$-vertex triangle-free graph. Then $G$ has at most $\left\lfloor n^2/4\right\rfloor$ edges.

Theorems & Definitions (9)

  • Theorem 1: Mantel, Man
  • Theorem 2: Lovász and Simonovits, Lov
  • Theorem 3
  • Proposition 1
  • Theorem 4: Harris, Har
  • Lemma 1
  • proof
  • Claim 1
  • proof