On the singular pebbling number of a graph

TL;DR

This paper introduces the singular pebbling number, a new graph parameter related to pebbling numbers, and establishes its values for various classes of graphs, showing it often equals the pebbling number.

Contribution

It defines the singular pebbling number and proves its equality to the pebbling number for most graphs, providing exact values for specific small graphs.

Findings

Singular pebbling number equals pebbling number for graphs with 3 or more vertices.

For disconnected graphs on two vertices, the singular pebbling number equals the pebbling number.

Exact singular pebbling numbers are determined for the graphs K_1 and K_2.

Abstract

In this paper, we define a new parameter of a graph as a spin-off of the pebbling number (which is the smallest $t$ such that every supply of $t$ pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest $t$ such that a player can be given any configuration of at least $t$ pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal to its pebbling number, that the singular pebbling number of the disconnected graph on two vertices is equal to its pebbling number, and we find the singular pebbling numbers of the two remaining graphs, $K_1$ and $K_2$, which are not equal to their pebbling numbers.