A graph-theoretic approach to Wilf's conjecture

TL;DR

This paper uses graph theory to extend the verification of Wilf's conjecture for numerical semigroups, proving it holds when the number of primitive elements is at least one-third of the multiplicity, covering most cases up to genus 45.

Contribution

The paper introduces a graph-theoretic method to verify Wilf's conjecture for cases where the primitive elements are at least one-third of the multiplicity, expanding previous results.

Findings

Wilf's conjecture holds for |P| ≥ m/3 in numerical semigroups.

This case covers over 99.999% of semigroups with genus ≤ 45.

The approach extends previous partial verifications using graph theory.

Abstract

Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}) and conductor c = max(N \ S) + 1. Let P be the set of primitive elements of S, and let L be the set of elements of S which are smaller than c. A longstand-ing open question by Wilf in 1978 asks whether the inequality |P||L| $\ge$ c always holds. Among many partial results, Wilf's conjecture has been shown to hold in case |P| $\ge$ m/2 by Sammartano in 2012. Using graph theory in an essential way, we extend the verification of Wilf's conjecture to the case |P| $\ge$ m/3. This case covers more than 99.999% of numerical semigroups of genus g $\le$ 45.