A Solution for the Open Abelian Sandpile Problem of Distributing k Items in N Vertices, where k = N

TL;DR

This paper presents a closed-form solution to classify initial distributions in the open Abelian Sandpile problem where the number of items equals the number of vertices, using an algorithm and modulus classification.

Contribution

It introduces a novel closed solution and an enumeration algorithm for successful initial distributions in the specific case where k equals N.

Findings

Successful distributions are classified by a common modulus.

An enumeration algorithm for initial distributions is provided.

The solution advances understanding of the Abelian Sandpile problem.

Abstract

This paper outlines a closed solution to an open problem in Graph Theory concerning the classification of the successful initial distributions of k items in N vertices, where $k = N$, that lead to the terminal set $N_k = \{ n_i\} $, where $n_i=1$ and $i=1,2,3,...,k$. First, each successful initial distribution is enumerated using an algorithm. The closed solution classifies the terminal set in terms of its modulus, and proves that each successful initial distribution can be classified by the same modulus.