The burning number conjecture holds for trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916934275039232 |
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| author | Ning, Jiajun Jin, Xian'an Zhang, Meiqiao |
| author_facet | Ning, Jiajun Jin, Xian'an Zhang, Meiqiao |
| contents | Inspired by the spread of information in social networks and graph-theoretic processes such as Firefighting and graph cleaning, Bonato, Janssen and Roshanbin introduced in 2016 the burning number $b(G)$ of any finite graph $G$. They conjectured that $b(G)\le \lceil n^\frac{1}{2}\rceil$ holds for all connected graphs $G$ of order $n$, and observed that it suffices to prove the conjecture for all trees. In 2024, Murakami confirmed the conjecture for trees without degree-2 vertices. In this paper, we prove that for all trees $T$ of order $n$ with $n_2$ degree-2 vertices, $$b(T)\le \left\lceil \left(n+n_2-\left\lceil\sqrt{n+n_2+0.25}-1.5\right\rceil\right)^{\frac{1}{2}}\right\rceil.$$ Hence, the conjecture holds for all trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_03144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The burning number conjecture holds for trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices Ning, Jiajun Jin, Xian'an Zhang, Meiqiao Combinatorics 05C05, 05C57 Inspired by the spread of information in social networks and graph-theoretic processes such as Firefighting and graph cleaning, Bonato, Janssen and Roshanbin introduced in 2016 the burning number $b(G)$ of any finite graph $G$. They conjectured that $b(G)\le \lceil n^\frac{1}{2}\rceil$ holds for all connected graphs $G$ of order $n$, and observed that it suffices to prove the conjecture for all trees. In 2024, Murakami confirmed the conjecture for trees without degree-2 vertices. In this paper, we prove that for all trees $T$ of order $n$ with $n_2$ degree-2 vertices, $$b(T)\le \left\lceil \left(n+n_2-\left\lceil\sqrt{n+n_2+0.25}-1.5\right\rceil\right)^{\frac{1}{2}}\right\rceil.$$ Hence, the conjecture holds for all trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices. |
| title | The burning number conjecture holds for trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices |
| topic | Combinatorics 05C05, 05C57 |
| url | https://arxiv.org/abs/2509.03144 |