Matching Numbers and Dimension of Edge Ideals

Abstract

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) and ind-match(G) the matching number, minimum matching number and induced matching number of G, respectively. Let $$K[V(G)] = K[x_{1}, \ldots , x_{n}]$$ denote the polynomial ring over a field K and $$I(G) \subset K[V(G)]$$ the edge ideal of G. The relationship between these graph-theoretic invariants and ring-theoretic invariants of the quotient ring K[V(G)]/I(G) has been studied. In the present paper, we study the relationship between match(G), min-match(G), ind-match(G) and $${\hbox {dim}}K[V(G)]/I(G)$$ .