Stacking and clearing in graph pebbling
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| Format: | Preprint |
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2026
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| _version_ | 1866913059006578688 |
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| author | Csernák, Tamás Soukup, Lajos |
| author_facet | Csernák, Tamás Soukup, Lajos |
| contents | Suppose that pebbles are distributed on the vertices of a graph G. A pebbling step along an edge uv removes two pebbles from u and places one pebble on v.
We introduce two new graph parameters:
stack(G): the least integer t such that every configuration with t pebbles can be transformed, by a finite sequence of pebbling steps, into a configuration with all pebbles on a single vertex.
clear(G): defined analogously, but requiring that from every configuration with t pebbles, all but one pebble can be removed.
We prove that stack(G) is defined exactly for connected graphs, and that clear(G) is defined exactly for connected non-bipartite graphs. We also establish general upper bounds for these parameters; in particular,
stack(G), clear(G) <= 2 |V(G)| 2^diam(G),
where diam(G) denotes the diameter of G.
Among our exact results are the equalities
stack(K_n) = clear(K_n) = n + 1,
stack(K_{m,n}) = 3 max{m,n} + 1,
stack(P_n) = 2^n - 1.
We also establish general lower bounds in terms of the independence number and odd closed walks.
For cycles, the situation is more delicate. We prove the lower bounds
stack(C_{2n}) >= 2^{n+1} - 1,
clear(C_{2n+1}) >= 3 * 2^n - 2,
and formulate the Almost Stacked Hypothesis, motivated by Sjostrand's cover pebbling theorem. Assuming this hypothesis, we obtain
stack(C_{2n}) = 2^{n+1} - 1,
clear(C_{2n+1}) = 3 * 2^n - 2.
At present, we do not have a conjecture for the exact value of stack(C_{2n+1}). Finally, computational evidence leads us to a conjectural closed formula for the stacking number of a tree in terms of the distances and degrees of the vertices relative to a chosen root. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22341 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stacking and clearing in graph pebbling Csernák, Tamás Soukup, Lajos Combinatorics 05C57, 05C05, 05C38 Suppose that pebbles are distributed on the vertices of a graph G. A pebbling step along an edge uv removes two pebbles from u and places one pebble on v. We introduce two new graph parameters: stack(G): the least integer t such that every configuration with t pebbles can be transformed, by a finite sequence of pebbling steps, into a configuration with all pebbles on a single vertex. clear(G): defined analogously, but requiring that from every configuration with t pebbles, all but one pebble can be removed. We prove that stack(G) is defined exactly for connected graphs, and that clear(G) is defined exactly for connected non-bipartite graphs. We also establish general upper bounds for these parameters; in particular, stack(G), clear(G) <= 2 |V(G)| 2^diam(G), where diam(G) denotes the diameter of G. Among our exact results are the equalities stack(K_n) = clear(K_n) = n + 1, stack(K_{m,n}) = 3 max{m,n} + 1, stack(P_n) = 2^n - 1. We also establish general lower bounds in terms of the independence number and odd closed walks. For cycles, the situation is more delicate. We prove the lower bounds stack(C_{2n}) >= 2^{n+1} - 1, clear(C_{2n+1}) >= 3 * 2^n - 2, and formulate the Almost Stacked Hypothesis, motivated by Sjostrand's cover pebbling theorem. Assuming this hypothesis, we obtain stack(C_{2n}) = 2^{n+1} - 1, clear(C_{2n+1}) = 3 * 2^n - 2. At present, we do not have a conjecture for the exact value of stack(C_{2n+1}). Finally, computational evidence leads us to a conjectural closed formula for the stacking number of a tree in terms of the distances and degrees of the vertices relative to a chosen root. |
| title | Stacking and clearing in graph pebbling |
| topic | Combinatorics 05C57, 05C05, 05C38 |
| url | https://arxiv.org/abs/2604.22341 |