Rubbling and optimal rubbling of graphs

Christopher Belford, Nándor Sieben

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


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 languageEnglish (US)
Pages (from-to)3436-3446
Number of pages11
JournalDiscrete Mathematics
Issue number10
StatePublished - May 28 2009


  • Optimal pebbling
  • Pebbling
  • Rubbling

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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