Standard Graded Vertex Cover Algebras, Cycles and Leaves

i∈F ci ≥ k for all F ∈ F(∆). If c is a (0, 1)-vector, then c may be identified with the subset C = {i ∈ [n] : ci 6= 0} of [n]. It is clear that c is a 1-cover if and only if C is a vertex cover of ∆ in the classical sense, that is, C ∩ F 6= ∅ for all F ∈ F(∆). Let S = K[x1, . . . , xn] be a polynomial ring in n variables over a field K. Let Ak(∆) denote the K-vector space generated by all monomials x c1 1 · · ·x cn n t such that (c1, . . . , cn) ∈ N n is a k-cover of ∆, where t is a new variable. Then