Abstract
A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3436-3446 |
| Number of pages | 11 |
| Journal | Discrete Mathematics |
| Volume | 309 |
| Issue number | 10 |
| DOIs | |
| State | Published - May 28 2009 |
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Keywords
- Optimal pebbling
- Pebbling
- Rubbling
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
Cite this
Rubbling and optimal rubbling of graphs. / Belford, Christopher; Sieben, Nandor.
In: Discrete Mathematics, Vol. 309, No. 10, 28.05.2009, p. 3436-3446.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Rubbling and optimal rubbling of graphs
AU - Belford, Christopher
AU - Sieben, Nandor
PY - 2009/5/28
Y1 - 2009/5/28
N2 - A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.
AB - A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs and we show that Graham's conjecture does not hold for rubbling numbers.
KW - Optimal pebbling
KW - Pebbling
KW - Rubbling
UR - http://www.scopus.com/inward/record.url?scp=67349112995&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=67349112995&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2008.09.035
DO - 10.1016/j.disc.2008.09.035
M3 - Article
VL - 309
SP - 3436
EP - 3446
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 10
ER -