An Improved Algorithm for Counting Graphical Degree Sequences

TL;DR

This paper introduces a faster algorithm for counting zero-free graphical degree sequences of length n, improving computational efficiency and enabling estimation of their asymptotic behavior through simulations.

Contribution

It presents an improved, more efficient algorithm for computing the number of graphical degree sequences, adaptable for all lengths up to n, with theoretical and experimental validation.

Findings

Algorithm is about 10 times faster than previous methods.

New techniques facilitate asymptotic estimation of D(n).

Method can be applied to other related counting functions.

Abstract

We present an improved version of a previous efficient algorithm that computes the number $D(n)$ of zero-free graphical degree sequences of length $n$. A main ingredient of the improvement lies in a more efficient way to compute the function $P(N,k,l,s)$ of Barnes and Savage. We further show that the algorithm can be easily adapted to compute the $D(i)$ values for all $i\le n$ in a single run. Theoretical analysis shows that the new algorithm to compute all $D(i)$ values for $i\le n$ is a constant times faster than the previous algorithm to compute a single $D(n)$. Experimental evaluations show that the constant of improvement is about 10. We also perform simulations to estimate the asymptotic order of $D(n)$ by generating uniform random samples from the set of $E(n)$ integer partitions of fixed length $n$ with even sum and largest part less than $n$ and computing the proportion of them that are graphical degree sequences. The known numerical results of $D(n)$ for $n\le 290$ together with the known bounds of $D(n)$ and simulation results allow us to make an informed guess about its unknown asymptotic order. The techniques for the improved algorithm can be applied to compute other similar functions that count the number of graphical degree sequences of various classes of graphs of order $n$ and that all involve the function $P(N,k,l,s)$.