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.
graph pebbling, Lemke graph, two-pebbling, odd-two-pebbling
Repository citation: Cusack, Charles A.; Bekmetjev, Airat; and Powers, Mark, "Two-pebbling and Odd-two-pebbling are Not Equivalent" (2019). Faculty Publications. Paper 1485.
Published in: Discrete Mathematics, Volume 342, Issue 3, March 1, 2019, pages 777-783. Copyright © 2019 Elsevier.