Dirac's theorem on chordal graphs and Alexander duality
European Journal of Combinatorics
Characteristic-independence of Betti numbers of graph ideals
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Simplicial cycles and the computation of simplicial trees
Journal of Symbolic Computation
Whiskers and sequentially Cohen--Macaulay graphs
Journal of Combinatorial Theory Series A
Regularity, depth and arithmetic rank of bipartite edge ideals
Journal of Algebraic Combinatorics: An International Journal
Path ideals of rooted trees and their graded Betti numbers
Journal of Combinatorial Theory Series A
Koszulness, Krull dimension, and other properties of graph-related algebras
Journal of Algebraic Combinatorics: An International Journal
Algorithmic complexity of finding cross-cycles in flag complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
Projective dimension, graph domination parameters, and independence complex homology
Journal of Combinatorial Theory Series A
Mengerian quasi-graphical families and clutters
European Journal of Combinatorics
Journal of Algebraic Combinatorics: An International Journal
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We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph 驴 appears within the resolution of its edge ideal 驴(驴). We discuss when recursive formulas to compute the graded Betti numbers of 驴(驴) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405---425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph 驴 (and thus, for any simple graph), we give a lower bound for the regularity of 驴(驴) via combinatorial information describing 驴 and an upper bound for the regularity when 驴=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When 驴 is a triangulated hypergraph, we explicitly compute the regularity of 驴(驴) and show that the graded Betti numbers of 驴(驴) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.