Accessible set systems and a conjecture on Cohen-Macaulay binomialedge ideals

• Autores: Davide Bolognini, Antonio Macchia, Giancarlo Rinaldo, Francesco Strazzanti
• Localización: Discrete Mathematics Days 2022 / Luis Felipe Tabera Alonso (ed. lit.), 2022, ISBN 978-84-19024-02-2, págs. 49-53
• Idioma: inglés
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• Resumen

  • Binomial edge ideals are generated by the 2-minors of a generic matrix with two rowswhose column indices correspond to the edges of a graph. They naturally generalize theideals of 2-minors and appear in Algebraic Statistics. Many algebraic properties of binomialedge ideals can be studied through the cut sets of the associated graph, which are specialsets of vertices whose removal disconnects the graph. A recent conjecture by some of theauthors states that Cohen-Macaulay binomial edge ideals are exactly the binomial edgeideals JG whose graph is accessible, i.e., JG is unmixed and the collection of cut sets ofG is an accessible set system. Exploiting the structure of accessible graphs, we providefurther evidence in support of this conjecture, proving that it holds for all graphs with up 15vertices. The computations for graphs with 13, 14 and 15 vertices could not be completedwith a naive brute force approach.