Hassan Haghighi, Siamak Yassemi and Rahim Zaare-Nahandi: Bipartite $S_2$ graphs are Cohen-Macaulay, p.125-132

Abstract:

In this paper we show that if the Stanley-Reisner ring of the simplicial complex of independent sets of a bipartite graph $G$ satisfies Serre's condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi carries over this family of bipartite graphs. We check that the equivalence of Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs and we classify cyclic graphs with respect to the condition $S_2$.

Key Words: Bipartite graph, Cohen-Macaulay graph, Serre's condition, chordal graph

2000 Mathematics Subject Classification: Primary: 13H10,
Secondary: 05C75.

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