Most probably trangle-free graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917295786295296 |
|---|---|
| author | Bai, Yuhang Katona, Gyula O. H. Yang, Zixuan |
| author_facet | Bai, Yuhang Katona, Gyula O. H. Yang, Zixuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22782 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Most probably trangle-free graphs Bai, Yuhang Katona, Gyula O. H. Yang, Zixuan Combinatorics 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. |
| title | Most probably trangle-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.22782 |