Connections with Combinatorics

Abstract

This chapter investigates three special cases of monomial ideals that are important for graph theory and combinatorics: the edge ideal of a simple graph and the Stanley-Reisner and facet ideals of a simplicial complex. Each of these cases is a monomial ideal that is “square free”. These ideals are treated, in general, in Section 4.1. Graphs and their edge ideals are introduced in Section 4.2, and the decompositions of edge ideals are described in Section 4.3. Simplicial complexes and their Stanley-Reisner ideals are presented in Section 4.4, and the decompositions of Stanley-Reisner ideals are described in Section 4.5. This includes, as a consequence, a method for finding decompositions of arbitrary square-free monomial ideals. Section 4.6 treats the facet ideals associated to simplicial complexes, and their m-irreducible decompositions. The chapter ends in Section 4.7 with an exploration of Alexander duality, a process that transforms monomial generating sequences to m-irreducible decompositions, and vice versa.