Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

Gyula Y. Katona, Nandor Sieben

Research output: Contribution to journalArticle

3 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 each 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 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 give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

Original languageEnglish (US)
Pages (from-to)535-551
Number of pages17
JournalGraphs and Combinatorics
Issue number3
StatePublished - May 2013



  • Pebbling
  • Rubbling

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

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