Simplicial trees are sequentially Cohen–Macaulay

Abstract. Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 , . . . , x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

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“…Theorem 3.2 also generalizes the one-dimensional case of the work of Faridi on simplicial forests [4].…”

Section: Theorem 12 (Theorem 32) All Chordal Graphs Are Sequentialmentioning

confidence: 69%

“…If it were, then there would be a minimal vertex cover m that divided it. But then x 2 x 3 x 5 x 6 x 8 x 2r+1 divides m since x 1 , x 4 , and x 7 are missing, and m is a cover. If 2r + 1 > 9, then to cover the remaining 2r − 9 edges not covered, we need at least r − 4 vertices.…”

Section: We Show That J Does Not Have a Linear Resolution This Implimentioning

confidence: 99%

Sequentially Cohen-Macaulay edge ideals

Francisco

Tuyl

2007

Proc. Amer. Math. Soc.

100

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“…For this reason I is called the edge ideal of C and is denoted I = I (C). Edge ideals of clutters are also called facet ideals [8] because S 1 , . .…”

Section: Clutters With the Free Vertex Property Are Shellablementioning

confidence: 99%

Shellable graphs and sequentially Cohen–Macaulay bipartite graphs

Tuyl

Villarreal

2008

Journal of Combinatorial Theory, Series A

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Associated to a simple undirected graph G is a simplicial complex Δ G whose faces correspond to the independent sets of G. We call a graph G shellable if Δ G is a shellable simplicial complex in the nonpure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give a recursive procedure to verify if a bipartite graph is shellable. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.

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“…Let G be a simple graph and let S ⊂ V G . Suppose that G \ S is a chordal graph or a five-cycle C 5 . Then G ∪ W (S) is a sequentially Cohen-Macaulay graph.…”

Section: Introductionmentioning

confidence: 99%

Whiskers and sequentially Cohen–Macaulay graphs

Francisco

Harbourne

2008

Journal of Combinatorial Theory, Series A

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We investigate how to modify a simple graph G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G ∪ W (S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G ∪ W (S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G \ S is not sequentially Cohen-Macaulay, then G ∪ W (S) is not a sequentially CohenMacaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.

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Simplicial trees are sequentially Cohen–Macaulay

Cited by 72 publications

References 13 publications

Sequentially Cohen-Macaulay edge ideals

Sequentially Cohen-Macaulay edge ideals

Shellable graphs and sequentially Cohen–Macaulay bipartite graphs

Whiskers and sequentially Cohen–Macaulay graphs

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