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.
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. A
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. A