Factor critical graphs and irreducible decomposition of powers of edge ideals

A graph G is said to be factor-critical if G−v has a perfect matching for each vertex v. We study the relation between factor criticality and irreducible decomposition, which are usually considered in the theory of commutative algebra. In particular, we give necessary and sufficient conditions for a graph to be factor critical in terms of irreducible decomposition of monomial ideals. We also investigate the relations between irreducible decomposition of monomial ideals and matching theory.