Shellable graphs and sequentially Cohen-Macaulay bipartite graphs |
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Authors: | Adam Van Tuyl Rafael H Villarreal |
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Institution: | a Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada b Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Mexico City, DF, Mexico |
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Abstract: | Associated to a simple undirected graph G is a simplicial complex ΔG whose faces correspond to the independent sets of G. We call a graph G shellable if ΔG is a shellable simplicial complex in the non-pure sense of Björner-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 a 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. |
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Keywords: | Shellable complex Sequentially Cohen-Macaulay Edge ideals Bipartite and chordal graphs Totally balanced clutter |
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