Achievable Burning Densities of Growing Grids
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909995869667328 |
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| author | Barrett, Jordan Gunderson, Karen Nir, JD Pralat, Pawel |
| author_facet | Barrett, Jordan Gunderson, Karen Nir, JD Pralat, Pawel |
| contents | Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set of achievable burning densities for functions of the form $f(n)=\lceil cn^α\rceil$, where $α\ge 1$ and $c>0$. We show that for $α=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<α<3/2$, every density in $[0,1]$ is achievable, and for $α=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14151 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Achievable Burning Densities of Growing Grids Barrett, Jordan Gunderson, Karen Nir, JD Pralat, Pawel Combinatorics Discrete Mathematics Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set of achievable burning densities for functions of the form $f(n)=\lceil cn^α\rceil$, where $α\ge 1$ and $c>0$. We show that for $α=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<α<3/2$, every density in $[0,1]$ is achievable, and for $α=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$. |
| title | Achievable Burning Densities of Growing Grids |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2601.14151 |