Pebbling in powers of paths

Abstract

The t-fold pebbling number, πt(G), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least t pebbles on any specified vertex via pebbling moves. It has been conjectured that the pebbling numbers of pyramid-free chordal graphs can be calculated in polynomial time.The kth power G(k) of the graph G is obtained from G by adding an edge between any two vertices of distance at most k from each other. The kth power of the path Pn on n vertices is an important class of pyramid-free chordal graphs, and is a stepping stone to the more general class of k-paths and the still more general class of interval graphs. Pachter, Snevily, and Voxman (1995) calculated π(Pn(2)), Kim (2004) calculated π(Pn(3)), and Kim and Kim (2010) calculated π(Pn(4)). In this paper we calculate πt(Pn(k)) for all n, k, and t.For a function D:V(G)→N, the D-pebbling number, π(G,D), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least D(v) pebbles on each vertex v via pebbling moves. It has been conjectured that π(G,D)≤π|D|(G) for all G and D. We make the stronger conjecture that every G and D satisfies π(G,D)≤π|D|(G)−(s(D)−1), where s(D) counts the number of vertices v with D(v)>0. We prove that trees and Pn(k), for all n and k, satisfy the stronger conjecture.The pebbling exponenteπ(G) of a graph G was defined by Pachter et al., to be the minimum k for which π(G(k))=n(G(k)). Of course, eπ(G)≤diam(G), and Czygrinow, Hurlbert, Kierstead, and Trotter (2002) proved that almost all graphs G have eπ(G)=1. Lourdusamy and Mathivanan (2015) proved several results on πt(Cn2), and Hurlbert (2017) proved an asymptotically tight formula for eπ(Cn). Our formula for πt(Pn(k)) allows us to compute eπ(Pn) within additively narrow bounds.