A note on the optimal rubbling in ladders and prisms

TL;DR

This paper investigates the optimal rubbling number in specific graph classes, providing concise proofs for cycles, paths, ladders, prisms, and Mobius-ladders, advancing understanding of pebbling dynamics in these structures.

Contribution

It offers new, simplified proofs for the rubbling number and optimal rubbling number in various graph families, expanding theoretical knowledge in graph pebbling.

Findings

Rubbling number of cycles determined

Optimal rubbling number of paths and cycles established

Rubbling properties of ladders, prisms, and Mobius-ladders analyzed

Abstract

A pebbling move on a graph G consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u and v adjacent to a vertex w, and one pebble is added on w. The optimal rubbling number of a graph G is the smallest number m, such that one pebble can be moved to every given vertex from some pebble distribution of m pebbles by a sequence of rubbling moves. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, ladders, prisms and Mobius-ladders.