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The Middle Stair of Parallel Chip-Firing

Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.

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combinatoricsAug 4, 2026Significance 12/100Registry: expert verified

The Middle Stair of Parallel Chip-Firing

Prior state unknown→proved

Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.

Source authenticatedThe Middle Stair of Parallel Chip-Firing →
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Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.

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