Cohen-Macaulay graphs with large girth

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

doi:10.48550/arxiv.1204.5561

Cited by 2 publications

(3 citation statements)

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“…Remark 21. We note that Theorem 20 is also discovered independently by Hoang, Minh and Trung [8] (compare to Theorem 3.1 in [8]).…”

Section: Corollary 16 If a Bipartite Graph G Is Vertex-decomposable T...supporting

confidence: 61%

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

Bıyıkoğlu¹,

Civan²

2012

Preprint

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for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C 4 , C 5 )-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C 4 , C 5 , C 7 )-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity reg(G) of such graphs, and show that reg(G) = im(G) whenever G is a (C 4 , C 5 )-free vertex-decomposable graph, where im(G) is the induced matching number of G. Furthermore, we prove that H must be a codismantlable graph if im(H) = reg(H) = m(H), where m(H) is the matching number of H. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family H n (n ≥ 4) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and reg(H n ) = im(H n ) if n ≥ 6. This answers a recent question of Mahmoudi, et al [12].

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“…Remark 21. We note that Theorem 20 is also discovered independently by Hoang, Minh and Trung [8] (compare to Theorem 3.1 in [8]).…”

Section: Corollary 16 If a Bipartite Graph G Is Vertex-decomposable T...supporting

confidence: 61%

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

Bıyıkoğlu¹,

Civan²

2012

Preprint

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“…Remark 5. 16. We note that a graph G contains no g-pair if and only if every edge of G is contained in an induced 2K 2 of G, that is, its complement G is a 4-cycled graph (see [3] for details).…”

Section: ýñmentioning

confidence: 99%

“…Finally, the t-contraction of tb, y d u in tptpG; dq; acq yields a graph isomorphic to fppG´tb, cuq; daqYK 2 , where the isolated edge is induced by w ac and w by d (see to Figure 9). Our next aim is to prove the equality regpGq " vimpGq when G is a well-covered blockcactus graph [33] that in turn includes any Cohen-Macaulay graph with girth at least five (see also [5,16]). We recall that a graph G is called a block-cactus graph, if each of its blocks is a clique or a cycle.…”

Section: ýñmentioning

confidence: 99%

Castelnuovo-Mumford Regularity of Graphs

Bıyıkoğlu¹,

Civan

2018

Combinatorica

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We present new combinatorial insights into the calculation of (Castelnuovo-Mumford) regularity of graphs. We first show that the regularity of any graph can be reformulated as a generalized induced matching problem. On that direction, we introduce the notion of a prime graph by calling a connected graph G as a prime graph over a field k, if reg k pG´xq ă reg k pGq for any vertex x P V pGq. We exhibit some structural properties of prime graphs that enables us to compute the regularity in specific hereditary graph classes. In particular, we prove that regpGq ď ∆pGq impGq holds for any graph G, and in the case of claw-free graphs, we verify that this bound can be strengthened by showing that regpGq ď 2 impGq, where impGq is the induced matching number of G. By analysing the effect of Lozin transformations on graphs, we narrow the search of prime graphs into bipartite graphs having sufficiently large girth with maximum degree at most three, and show that the regularity of bipartite graphs G with such constraints is bounded above by 2 impGq`1. Moreover, we prove that any non-trivial Lozin operation preserves the primeness of a graph that enables us to generate new prime graphs from the existing ones.We introduce a new graph invariant, the virtual induced matching number vimpGq satisfying impGq ď vimpGq ď regpGq for any graph G that results from the effect of edgecontractions and vertex-expansions both on graphs and the independence complexes of graphs to the regularity. In particular, we verify the equality regpGq " vimpGq for a graph class containing all Cohen-Macaulay graphs of girth at least five.Finally, we prove that there exist graphs satisfying regpGq " n and impGq " k for any two integers n ě k ě 1. The proof is based on a result of Januszkiewicz and Swiatkowski [18] accompanied with Lozin operations. We provide an upper bound on the regularity of any 2K 2 -free graph G in terms of the maximum privacy degree of G. In addition, if G is a prime 2K 2 -free graph, we show that regpGq ď δpGq`3 2 .In particular, Theorem 2.6 implies that IndpGq » IndpG´eq whenever the edge e is isolating. This brings the use of an operation, adding or removing an edge, on a graph without altering its homotopy type. We will follow [1] to write Addpx, y; wq (respectively Delpx, y; wq) to indicate that we add the edge e " xy to (resp. remove the edge e " xy from) the graph G, where w is the corresponding isolated vertex.Remark 2.7. In order to simplify the notation, we note that when we mention the homology, homotopy or a suspension of a graph, we mean that of its independence complex, so whenever it is appropriate, we drop Indp´q from our notation. Prime graphs and Prime FactorizationsAs we have already mentioned in Section 1, the notion of primeness brings a new strategy for the calculation of the regularity. Even if we express its definition in Section 1, there seems no harm for restating it in its greatest generality.Corollary 5.14. Let z be a non-isolated vertex of a graph G such that rA z , B z s is a t-pairing in G´N G rzs, then ...

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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 2 publications

References 11 publications

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

Castelnuovo-Mumford Regularity of Graphs

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