Chapter 9 deals with the algebraic aspects of Dirac’s theorem on chordal graphs and the classification problem for Cohen–Macaulay graphs. First the classification of bipartite Cohen–Macaulay graphs is given. Then unmixed graphs are characterized and we present the result which says that a bipartite graph is sequentially Cohen–Macaulay if and only if it is shellable. It follows the classification of Cohen–Macaulay chordal graphs. Finally the relationship between the Hilbert–Burch theorem and Dirac’s theorem on chordal graphs is explained.
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