Geodesic
A graph-theoretic approach to Wilf's conjecture
Shalom Eliahou1
1Université du Littoral Côte d'Opale
The electronic journal of combinatorics, Tome 27 (2020) no. 2
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Let $S \subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m = \min(S \setminus \{0\})$ and conductor $c=\max(\mathbb{N} \setminus S)+1$. 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 longstanding 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 $g \le 45$.
DOI : 10.37236/9106
Classification : 11D04, 20M14, 05C99
Mots-clés : Frobenius number, numerical semigroup

Shalom Eliahou  1

1 Université du Littoral Côte d'Opale 
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     author = {Shalom Eliahou},
     title = {A graph-theoretic approach to {Wilf's} conjecture},
     journal = {The electronic journal of combinatorics},
     year = {2020},
     volume = {27},
     number = {2},
     doi = {10.37236/9106},
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Shalom Eliahou. A graph-theoretic approach to Wilf's conjecture. The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/9106

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