Non-Stabilizing Parallel Chip-Firing Games
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arXiv
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2024
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| _version_ | 1866929472407601152 |
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| author | Ji, David Li, Michael Wang, Daniel |
| author_facet | Ji, David Li, Michael Wang, Daniel |
| contents | In 2010, Kominers and Kominers proved that any parallel chip-firing game on $G(V,\,E)$ with $|σ|\geq 4|E|-|V|$ chips stabilizes. Recently, Bu, Choi, and Xu made the bound exact: all games with $|σ|< |E|$ chips or $|σ|> 3|E|-|V|$ chips stabilize. Meanwhile, Levine found a "devil's staircase'' pattern in the plot of the activity of parallel chip-firing games against their density of chips. The stabilizing bound of Bu, Choi, and Xu corresponds to the top and bottom stairs of this staircase, in which the activity is 1 and 0, respectively. In this paper, we analyze the middle stair of the staircase, corresponding to activity $\frac{1}{2}$. We prove that all parallel chip-firing games with $2|E|-|V|< |σ|< 2|E|$ have period $T\neq 3,\,4$. In fact, this is exactly the range of $|σ|$ for which all games are non-stabilizing. We conjecture that all parallel chip-firing games with $2|E|-|V|< |σ|<2|E|$ have $T=2$ and thus activity $\frac{1}{2}$. This conjecture has been proven for trees by Bu, Choi, and Xu, cycles by Dall'asta, and complete graphs by Levine. We extend Levine's method of conjugate configurations to prove the conjecture on complete bipartite graphs $K_{a,a}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Stabilizing Parallel Chip-Firing Games Ji, David Li, Michael Wang, Daniel Combinatorics 05C57 68R10 In 2010, Kominers and Kominers proved that any parallel chip-firing game on $G(V,\,E)$ with $|σ|\geq 4|E|-|V|$ chips stabilizes. Recently, Bu, Choi, and Xu made the bound exact: all games with $|σ|< |E|$ chips or $|σ|> 3|E|-|V|$ chips stabilize. Meanwhile, Levine found a "devil's staircase'' pattern in the plot of the activity of parallel chip-firing games against their density of chips. The stabilizing bound of Bu, Choi, and Xu corresponds to the top and bottom stairs of this staircase, in which the activity is 1 and 0, respectively. In this paper, we analyze the middle stair of the staircase, corresponding to activity $\frac{1}{2}$. We prove that all parallel chip-firing games with $2|E|-|V|< |σ|< 2|E|$ have period $T\neq 3,\,4$. In fact, this is exactly the range of $|σ|$ for which all games are non-stabilizing. We conjecture that all parallel chip-firing games with $2|E|-|V|< |σ|<2|E|$ have $T=2$ and thus activity $\frac{1}{2}$. This conjecture has been proven for trees by Bu, Choi, and Xu, cycles by Dall'asta, and complete graphs by Levine. We extend Levine's method of conjugate configurations to prove the conjecture on complete bipartite graphs $K_{a,a}$. |
| title | Non-Stabilizing Parallel Chip-Firing Games |
| topic | Combinatorics 05C57 68R10 |
| url | https://arxiv.org/abs/2408.10508 |