Cohen–Macaulay binomial edge ideals and accessible graphs

• Published in: Journal of algebraic combinatorics Vol. 55; no. 4; pp. 1139 - 1170
• Main Authors: Bolognini, Davide, Macchia, Antonio, Strazzanti, Francesco
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022; Springer Nature B.V
• Subjects: Accessibility; Apexes; Combinatorial analysis; Combinatorics; Computer Science; Convex and Discrete Geometry; Graph theory; Graphs; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Traceable graphs; Accessible set systems; Cohen–Macaulay rings; 13C05; 05C25; Chordal graphs; Binomial edge ideals; 13H10
• ISSN: 0925-9899; 1572-9192; 1572-9192
• DOI: 10.1007/s10801-021-01088-w

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness .