The regularity of edge rings and matching numbers

Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq \mat(G) - 1$ if $G$ is bipartite.

Theorem 1. Let G be a finite connected simple graph. Then (a) reg K[G] ≤ mat(G), if G is non-bipartite and K[G] is normal; Lemma 2 stated below, which provides information on lattice points belonging to interiors of dilations of edge polytopes, is indispensable for the proof of Theorem 1.
Lemma 2. Suppose that (a 1 , . . . , a d ) ∈ Z d belongs to the interior q(P G \ ∂P G ) of the dilation qP G = {qα : α ∈ P G }, where q ≥ 1, of P G . Then each a i ≥ 1.
Proof. The facets of P G are described in [5,Theorem 1.7]. When W ⊂ [d], we write G W for the induced subgraph of G on W . Since K[G] is normal, it follows that P G possesses the integer decomposition property [4, p. 91]. In other words, each a ∈ qP G ∩ Z d is of the form a = (e i 1 + e j 1 ) + · · · + (e iq + e jq ), Let s ≥ 1 and s ′ ≥ 0. Let W j ∪ U j denote the vertex set of the bipartite graph In other words, no edge e ∈ E(G) satisfies e ⊂ T . Let G ′ denote the bipartite graph induced by T . Thus the edges of , 0), . . . , (0, . . . , 0, 1). Thus, since each (a 1 , . . . , a d ) ∈ qP G ∩ Z d satisfies a 1 = q, the assertion follows immediately. In the argument below, one will assume that G is not a star graph.
Let Let s ≥ 2. Let W j ∪ U j denote the vertex set of H j for which there is a ∈ W j with {a, i} ∈ E(G). Since G is not a star graph, one can assume that U 1 = ∅. Then T = W 2 ∪ · · · ∪ W s is independent and the bipartite graph induced by T is G [d]\(W 1 ∪U 1 ) . Hence T is acceptable ([5, p. 415]) and the hyperplane of R d defined by ξ∈W 1 x ξ = ξ ′ ∈U 1 x ξ ′ is a facet of qP G . Now, suppose that a i = 0. Since P G possesses the integer decomposition property, one has ξ∈W 1 a ξ = ξ ′ ∈U 1 a ξ ′ . Hence (a 1 , . . . , a d ) ∈ Z d cannot belong to q(P G \ ∂P G ). Thus a i > 0, as required.
We say that a finite subset L ⊂ E(G) is an edge cover of G if ∪ e∈L e = [d]. Let µ(G) denote the minimal cardinality of edge covers of G.
Proof. Since P G possesses the integer decomposition property, Lemma 2 guarantees that, if a ∈ q(P G \ ∂P G ) ∩ Z d , one has q ≥ µ(G).
Once Corollary 3 is established, to complete the proof of Theorem 1 is a routine job on computing regularity of normal toric rings.
Proof of Theorem 1. In each of the cases (a) and (b), since the edge ring K[G] is normal, it follows that the Hilbert function of K[G] coincides the Ehrhart function ([4, p. 100]) of the edge polytope P G , which says that the Hilbert series of K[G] is of the form (h 0 + h 1 λ + · · · + h s λ s )/(1 − λ) (dim P G )+1 with each h i ∈ Z and h s = 0. One has  In the bipartite case, let H be the bipartite graph of type (r+1, r+1). Its matching number is r + 1. Indeed, K[H] may be viewed as Hibi ring whose regularity is wellknown, see for example [2, Theorem 1.1]. At one vertex of H we attach a path graph of length 2(m − r) and call this new graph G. Then mat(G) = m + 1 and reg K[G] = reg K[H] = r, by the same reason as before.

Now, Corollary 3 guarantees that
These bounds for the regularity of K[G] are in general only valid, if K[G] is normal. Consider for example the graph G which consists of two disjoint triangles combined be a path of length ℓ. Then the defining ideal of K[G] is generated by a binomial of degree ℓ + 3, and hence reg K[G] = ℓ + 2, while the matching number of G is 2 + ⌈ℓ/2⌉.