Cohen–Macaulay graphs with large girth

Hoang, Do Trong; Minh, Nguyên Công; Trung, Trân Nam

doi:10.1142/s0219498815501121

Cited by 29 publications

(30 citation statements)

References 20 publications

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“…For well-covered graphs (and, more generally, for parity graphs [29]), the outcome of the game is independent of how the players move. Well-covered graphs also play an important role in commutative algebra, where they are typically referred to as unmixed graphs, see, e.g., [5,41,42,45,62,74,75]. The well-coveredness property of a graph is equivalent to the property that the simplicial complex of the independent sets of G is pure and generalizes the algebraically defined concept of a Cohen-Macaulay graph (see, e.g., [16]).…”

Section: Introductionmentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

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A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs.In this paper we obtain several structural and algorithmic results about localizable graphs. Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, C 4 -free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm. On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three. Furthermore, using localizable graphs we disprove a conjecture due to Zaare-Nahandi about kpartite well-covered graphs having all maximal cliques of size k. Our results unify and generalize several results from the literature.

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Section: Introductionmentioning

confidence: 99%

Graphs vertex-partitionable into strong cliques

Hujdurović

Milanič

Ries

2018

Discrete Mathematics

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“…ind-match {K 2 ,C 5 } (W (H n )) + ind-match {K 2 ,C 5 } (H)= ind-match {K 2 ,C 5 } (W (H n )) + 6, Therefore, ind-match {K 2 ,C 5 } (G n ) = ind-match {K 2 ,C 5 } (W (H n )) + 6 = ind-match(W (H n )) + 6 ≤ 2n + 6,where the second equality follows from the equalities ‡.SettingP (G n ) = V (W (H n ) \ x) ∪ {z, v, w} and C(G n ) = V (H) \ {u, v, w},we see that the graph G n belongs to the class PC and hence, by[14, Theorem 2.4], it is a vertex decomposable graph. Thus, for any n ≥ 13 and every s ≥ 1, we have2s + ind-match(G n ) − 1 ≤ 2s + ind-match(W (H n )) + 4 − 2s + ind-match {K 2 ,C 5 } (W (H n )) + 3 < 2s + ind-match {K 2 ,C 5 } (W (H n )) + 4 = 2s + ind-match {K 2 ,C 5 } (G n ) − 2 ≤ reg(I(G n ) s ) ≤ 2s + ind-match {K 2 ,C 5 } (G n ) − 2s + 2n + 6 − 1 < 2s + 5n 2s + cochord(G n ) − 1.…”

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confidence: 91%

Improved bounds for the regularity of edge ideals of graphs

Fakhari

Yassemi

2017

Collect. Math.

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Let G be a graph with edge ideal I(G). We recall the notions of min-match {K2,C5} (G) and ind-match {K2,C5} (G) from [23]. We show thatMoreover, we show thatwhere ord-match(G) denotes the ordered matching number of G. Finally, we construct infinitely many connected graphs which satisfy the following strict inequalities:2s + ind-match(G) − 1 < reg(I(G) s ) < 2s + cochord(G) − 1. This gives a positive answer to a question asked in [15].

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“…Even more fruitful interactions concern shellability, vertex decomposability and well-coveredness [6,7,11]. For example, every Cohen-Macaulay graph is well-covered, while each Gorenstein graph without isolated vertices belongs to W 2 [16]. Moreover, a triangle-free graph G is Gorenstein if and only if every non-trivial connected component of G belongs to W 2 [17].…”

Section: Introductionmentioning

confidence: 99%

1-well-covered graphs revisited

Levit

Mândrescu

2019

European Journal of Combinatorics

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A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975).A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (J. W. Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G ∈ W2 if and only if it is a 1-well-covered graph without isolated vertices.We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A ∈ Ind (G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs, which we further use in building 1-well-covered graphs by corona, join, and concatenation operations.

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Cohen–Macaulay graphs with large girth

Abstract: We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

Cited by 29 publications

References 20 publications

Graphs vertex-partitionable into strong cliques

Graphs vertex-partitionable into strong cliques

Improved bounds for the regularity of edge ideals of graphs

1-well-covered graphs revisited

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