A spectral Erdős-Rademacher theorem
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913382298288128 |
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| author | Li, Yongtao Lu, Lu Peng, Yuejian |
| author_facet | Li, Yongtao Lu, Lu Peng, Yuejian |
| contents | A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if $G$ is a graph on $n$ vertices with at least $\lfloor {n^2}/{4} \rfloor +1$ edges, then $G$ contains at least $\lfloor {n}/{2}\rfloor$ triangles. We prove a spectral version of Erdős--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A spectral Erdős-Rademacher theorem Li, Yongtao Lu, Lu Peng, Yuejian Combinatorics 05C50, 05C35 A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if $G$ is a graph on $n$ vertices with at least $\lfloor {n^2}/{4} \rfloor +1$ edges, then $G$ contains at least $\lfloor {n}/{2}\rfloor$ triangles. We prove a spectral version of Erdős--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs. |
| title | A spectral Erdős-Rademacher theorem |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2406.05609 |