101 research outputs found
Shellable graphs and sequentially Cohen-Macaulay bipartite graphs
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
A graph 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 , the graph 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 is Cohen-Macaulay if and only if it is pure shellable.
In this article we improve this result by showing that is Cohen-Macaulay if
and only if it is pure vertex decomposable. In addition, if denotes the
edge ideal of , we show that the Castelnuovo-Mumford regularity of
is equal to the maximum number of pairwise 3-disjoint edges of . This
improves Kummini's result on unmixed bipartite graphs.Comment: 11 page
Vertex decomposable graphs and obstructions to shellability
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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