3 research outputs found
Pebbling in Dense Graphs
A configuration of pebbles on the vertices of a graph is solvable if one can
place a pebble on any given root vertex via a sequence of pebbling steps. The
pebbling number of a graph G is the minimum number pi(G) so that every
configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling
number equals its number of vertices. A function is a pebbling threshold for a
sequence of graphs if a randomly chosen configuration of asymptotically more
pebbles is almost surely solvable, while one of asymptotically fewer pebbles is
almost surely not. Here we prove that graphs on n>=9 vertices having minimum
degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336
vertices in each part having minimum degree at least floor(m/2)+1. Both bounds
are best possible. In addition, we prove that the pebbling threshold of graphs
with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when
d is proportional to n.Comment: 10 page