101 research outputs found

    Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

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    Associated to a simple undirected graph G is a simplicial complex whose faces correspond to the independent sets of G. We call a graph G shellable if this simplicial complex is a shellable simplicial complex in the non-pure sense of Bjorner-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 an 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.Comment: 16 pages; more detail added to some proofs; Corollary 2.10 was been clarified; the beginning of Section 4 has been rewritten; references updated; to appear in J. Combin. Theory, Ser.

    Vertex decomposability and regularity of very well-covered graphs

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    A graph GG is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to ∣V(G)∣|V(G)|, the graph GG is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph GG is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that GG is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if I(G)I(G) denotes the edge ideal of GG, we show that the Castelnuovo-Mumford regularity of R/I(G)R/I(G) is equal to the maximum number of pairwise 3-disjoint edges of GG. This improves Kummini's result on unmixed bipartite graphs.Comment: 11 page

    Vertex decomposable graphs and obstructions to shellability

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    Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and additional references. v3: minor corrections for publicatio
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