Spectral extremal results for triangle-free graphs with chromatic number at least four
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| Format: | Preprint |
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2026
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| _version_ | 1866917495537926144 |
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| author | Zhu, Yinfen Lin, Huiqiu |
| author_facet | Zhu, Yinfen Lin, Huiqiu |
| contents | A graph is called $F$-free if it does not contain a copy of $F$. Let $G(r,s)$ denote a $K_{r+1}$-free graph of order $n$ with chromatic number at least $s$ that maximizes the spectral radius. Nikiforov [Linear Algebra Appl., 2007] proved the spectral Turán theorem, which implies that $G(r,s)$ is the $r$-partite Turán graph $T_{n,r}$ for $s\leq r$. Lin, Ning, and Wu [Combin. Probab. Comput., 2021] characterized the unique spectral extremal graph $G(2,3)$. This result was later extended by Li and Peng [SIAM J. Discrete Math., 2023] to all $s=r+1\geq 3$. In this paper, we push the characterization further by determining the unique extremal graph $G(2,4)$ for all sufficiently large $n$. Specifically, we show that $G(2,4)$ is precisely a blow-up of the Grötzsch graph. Interestingly, under the same conditions, $G(2,4)$ also coincides with the unique edge-extremal graph identified by Ren, Wang, Wang, and Yang [arXiv:2404.07486v2]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14627 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral extremal results for triangle-free graphs with chromatic number at least four Zhu, Yinfen Lin, Huiqiu Combinatorics A graph is called $F$-free if it does not contain a copy of $F$. Let $G(r,s)$ denote a $K_{r+1}$-free graph of order $n$ with chromatic number at least $s$ that maximizes the spectral radius. Nikiforov [Linear Algebra Appl., 2007] proved the spectral Turán theorem, which implies that $G(r,s)$ is the $r$-partite Turán graph $T_{n,r}$ for $s\leq r$. Lin, Ning, and Wu [Combin. Probab. Comput., 2021] characterized the unique spectral extremal graph $G(2,3)$. This result was later extended by Li and Peng [SIAM J. Discrete Math., 2023] to all $s=r+1\geq 3$. In this paper, we push the characterization further by determining the unique extremal graph $G(2,4)$ for all sufficiently large $n$. Specifically, we show that $G(2,4)$ is precisely a blow-up of the Grötzsch graph. Interestingly, under the same conditions, $G(2,4)$ also coincides with the unique edge-extremal graph identified by Ren, Wang, Wang, and Yang [arXiv:2404.07486v2]. |
| title | Spectral extremal results for triangle-free graphs with chromatic number at least four |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.14627 |