This research was completed during a previous affiliation with the School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332.
Pebbling Graphs of Fixed Diameter
Article first published online: 21 FEB 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 75, Issue 4, pages 302–310, April 2014
How to Cite
Postle, L. (2014), Pebbling Graphs of Fixed Diameter. J. Graph Theory, 75: 302–310. doi: 10.1002/jgt.21736
- Issue published online: 15 JAN 2014
- Article first published online: 21 FEB 2013
- Manuscript Revised: 17 JAN 2013
- Manuscript Received: 26 FEB 2011
- graph pebbling;
- dominating sets
Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m-pebbling number of a graph G, , is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least m pebbles on v. When , it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and are integers, then , where denotes the size of the smallest distance k dominating set, that is the smallest subset of vertices such that every vertex is at most distance k from the set, and, . This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for . As a corollary, we prove that . Furthermore, we prove that if d is odd, then , which in the case of answers for odd d, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.