Graham's pebbling conjecture on Cartesian product of the middle graphs of even cycles

TL;DR

This paper investigates Graham's pebbling conjecture for the Cartesian product of middle graphs of even cycles, establishing the conjecture's validity in this specific case and calculating pebbling numbers for certain graphs.

Contribution

The paper proves Graham's pebbling conjecture for the middle graphs of some even cycles and computes pebbling numbers for specific graph classes.

Findings

Graham's conjecture holds for middle graphs of some even cycles.

Pebbling numbers are explicitly calculated for certain graphs.

The conjecture is verified in a new graph class.

Abstract

A pebbling move on a graph $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph $G$, denoted by $f(G)$, is the least integer $n$ such that, however $n$ pebbles are located on the vertices of $G$, we can move one pebble to any vertex by a sequence of pebbling moves. Let $M(G)$ be the middle graph of $G$. For any connected graphs $G$ and $H$, Graham conjectured that $f(G\times H)\leq f(G)f(H)$. In this paper, we give the pebbling number of some graphs and prove that Graham's conjecture holds for the middle graphs of some even cycles.