TY - JOUR
T1 - On pebbling threshold functions for graph sequences
AU - Czygrinow, Andrzej
AU - Eaton, Nancy
AU - Huribert, Glenn
AU - Kayll, P. Mark
N1 - Funding Information:
∗Corresponding author. E-mail addresses: andrzej@math.la.asu.edu (A. Czygrinow), eaton@math.uri.edu (N. Eaton), hurlbert@asu.edu (G. Hurlbert), kayll@charlo.math.umt.edu (P.M. Kayll). 1Visiting Professor. 2On sabbatical at Arizona State University. 3Partially supported by the Rocky Mountain Mathematics Consortium.
PY - 2002/3/28
Y1 - 2002/3/28
N2 - Given a connected graph G, and a distribution of / pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number /, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs 'S = (Gi,G2,...,G,...), where G has n vertices, is any function ta(n) such that almost all distributions of / pebbles are solvable when t>t0, and such that almost none are solvable when t<$to. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.
AB - Given a connected graph G, and a distribution of / pebbles to the vertices of G, a pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. For a particular vertex r, the distribution is r-solvable if it is possible to place a pebble on r after a finite number of pebbling steps. The distribution is solvable if it is r-solvable for every r. The pebbling number of G is the least number /, so that every distribution of t pebbles is solvable. In this paper we are not concerned with such an absolute guarantee but rather an almost sure guarantee. A threshold function for a sequence of graphs 'S = (Gi,G2,...,G,...), where G has n vertices, is any function ta(n) such that almost all distributions of / pebbles are solvable when t>t0, and such that almost none are solvable when t<$to. We give bounds on pebbling threshold functions for the sequences of cliques, stars, wheels, cubes, cycles and paths.
KW - Pebbling number
KW - S0012-365X(01)00163-7
KW - Threshold function
UR - http://www.scopus.com/inward/record.url?scp=31244437193&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=31244437193&partnerID=8YFLogxK
U2 - 10.1016/S0012-365X(01)00163-7
DO - 10.1016/S0012-365X(01)00163-7
M3 - Article
AN - SCOPUS:31244437193
VL - 247
SP - 93
EP - 105
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 1-3
ER -