Induced matching numbers of finite graphs and edge ideals

Abstract

Let G be a finite simple graph on the vertex set V(G)={x1,…,xn} and I(G)⊂K[V(G)] its edge ideal, where K[V(G)] is the polynomial ring in x1,…,xn over a field K with each deg⁡xi=1 and where I(G) is generated by those squarefree quadratic monomials xixj for which {xi,xj} is an edge of G. In the present paper, given integers 1≤a≤r and s≥1, the existence of a finite connected simple graph G=G(a,r,s) with im(G)=a, reg(R/I(G))=r and deg⁡hK[V(G)]/I(G)(λ)=s, where im(G) is the induced matching number of G and where hK[V(G)]/I(G)(λ) is the h-polynomial of K[V(G)]/I(G).