Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

TL;DR

This paper investigates bounds on the rubbling and optimal rubbling numbers of graphs, providing new upper bounds and estimates based on graph diameter and size, advancing understanding of pebbling-related graph parameters.

Contribution

It introduces new bounds for rubbling and optimal rubbling numbers, including sharp bounds and estimates for specific graph classes based on diameter.

Findings

Upper bound for rubbling number of n-vertex, diameter d graphs

Estimates for maximum rubbling number of diameter 2 graphs

Sharp upper bound for the optimal rubbling number

Abstract

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $v$ and $w$ adjacent to a vertex $u$, and an extra pebble is added at vertex $u$. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number $m$ needed to guarantee that any vertex is reachable from any pebble distribution of $m$ pebbles. The optimal rubbling number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of $n$-vertex, diameter $d$ graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.