Document Type
Article
Publication Date
3-2019
Publication Source
Discrete Mathematics
Volume Number
342
Issue Number
3
First Page
777
Last Page
783
Publisher
Elsevier
ISSN
0012-365X
Abstract
Let G be a connected graph. A configuration of pebbles assigns a nonnegative integer number of pebbles to each vertex of G. A move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if any vertex can get at least one pebble through a sequence of moves. The pebbling number of G, denoted π(G), is the smallest integer such that any configuration of π(G) pebbles on G is solvable. A graph has the two-pebbling property if after placing more than 2π(G) -- q pebbles on G, where q is the number of vertices with pebbles, there is a sequence of moves so that at least two pebbles can be placed on any vertex. A graph has the odd-two-pebbling property if after placing more than 2π(G) -- r pebbles on G, where r is the number of vertices with an odd number of pebbles, there is a sequence of moves so that at least two pebbles can be placed on any vertex. In this paper, we prove that the two-pebbling and odd-two-pebbling properties are not equivalent.
Keywords
graph pebbling, Lemke graph, two-pebbling, odd-two-pebbling
Recommended Citation
Repository citation: Cusack, Charles A.; Bekmetjev, Airat; and Powers, Mark, "Two-pebbling and Odd-two-pebbling are Not Equivalent" (2019). Faculty Publications. Paper 1485.
https://digitalcommons.hope.edu/faculty_publications/1485
Published in: Discrete Mathematics, Volume 342, Issue 3, March 1, 2019, pages 777-783. Copyright © 2019 Elsevier.
Comments
Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)
Charles A. Cusack, Airat Bekmetjev, Mark Powers, Two-pebbling and odd-two-pebbling are not equivalent, Discrete Mathematics, Volume 342, Issue 3, 2019, Pages 777-783, https://doi.org/10.1016/j.disc.2018.10.030.