On the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.


Introduction
Recently, Hibi and Tsuchiya [HT] showed that the Ehrhart ring of the stable set polytope of an odd cycle graph is Gorenstein if and only if the length of the cycle is less than or equal to 5. They used the fact that a cycle graph is t-perfect. On the other hand, Ohsugi and Hibi [OH] showed that the Ehrhart ring of the stable set polytope of a perfect graph is Gorenstein if and only if all maximal cliques have the same size.
Meanwhile, Sbihi and Uhry [SU] introduced the notion of h-perfect graphs as a common generalization of perfect and t-perfect graphs: a graph is hperfect if its stable set polytope is defined by the constraints nonnegativity of vertices and the constraints corresponding to cliques and odd cycles.
In this paper, we characterize Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph by using the trace of the canonical module of a Cohen-Macaulay ring. This result is a generalization of both results of Ohsugi-Hibi and Hibi-Tsuchiya.

Preliminaries
In this section, we establish notation and terminology. In this paper, all rings and algebras are assumed to be commutative with an identity element unless stated otherwise. Further, all graphs are finite simple graphs without loop. We denote the set of nonnegative integers, the set of integers, the set of rational numbers and the set of real numbers by N, Z, Q and R respectively. For a set X, we denote by #X the cardinality of X. For sets X and Y , we define X \ Y := {x ∈ X | x ∈ Y }. For nonempty sets X and Y , we denote the set of maps from X to Y by Y X . If X is a finite set, we identify R X with the Euclidean space R #X . For f , f 1 , f 2 ∈ R X and a ∈ R, we define maps f 1 ± f 2 and af by ( We denote the zero map X ∋ x → 0 ∈ R by 0. Let A be a subset of X. We define the characteristic function χ A ∈ R X by χ A (x) = 1 for x ∈ A and χ A (x) = 0 for x ∈ X \ A. For a nonempty subset X of R X , we denote by convX (resp. affX ) the convex hull (resp. affine span) of X .
A stable set of a graph G = (V, E) is a subset S of V with no two elements of S are adjacent. We treat the empty set as a stable set.
If HSTAB(G) = STAB(G), then we say that G is an h-perfect graph.
It is immediately seen that STAB(G) ⊂ HSTAB(G) for a general graph G. Note that HSTAB(G) is a convex polytope since it is bounded by (1) and (2).
We fix notation about Ehrhart rings. Let K be a field, X a finite set and P a rational convex polytope in R X , i.e., a convex polytope whose vertices are contained in Q X . Let −∞ be a new element with −∞ ∈ X and set We set deg T x = 0 for x ∈ X and deg T −∞ = 1. Then the Ehrhart ring of P over a field K is the N-graded subring where f | X is the restriction of f to X. We denote the Ehrhart ring of P over K by E K [P].
It is known that E K [P] is Noetherian. Therefore normal and Cohen-Macaulay by the criterion of Hochster [Hoc]. Further, by the description of the canonical module of a normal affine semigroup ring by Stanley [Sta,p. 82], we see the following.
, where relintP denotes the interior of P in the topological space affP.
We denote the ideal of the above lemma by ω E K [P] and call the canonical ideal of E K [P]. Let R be a Noetherian normal domain. Then the set of divisorial ideals Div(R) form a group by the operation I · J := R : Q(R) (R : Q(R) IJ) for I, J ∈ Div(R), where Q(R) is the quotient field of R. See e.g., [Fos, Chapter I] for details. We denote the n-th power of I ∈ Div(R) in this group by I (n) . Note that if R is a Cohen-Macaulay local or graded ring over a field with canonical module ω, then ω is isomorphic to a divisorial ideal. See e.g., [BH, Chapter 3] for details.

The trace of the canonical module and Gorenstein property
In this section, we give a criterion of Gorenstein property of the Ehrhart ring of HSTAB(G) of a graph G. As a consequence, we give a criterion of Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph. First we recall the following. Next we recall the following basic fact on the trace of an ideal. Note that if R is a domain and I is a divisorial ideal, then I −1 = I (−1) . Moreover, we recall the following.

Fact 3.3 ([HHS, Lemma 2.1])
Let R be a Cohen-Macaulay local or graded ring over a field with canonical module ω R . Then for p ∈ Spec(R), In particular, R is Gorenstein if and only if tr(ω R ) ∋ 1.
In the rest of this paper, we fix a graph G = (V, E). First we note the following.
(2) f + (K) ≤ 1 for any maximal clique K of G and (3) f + (C) ≤ #C−1 2 for any odd cycle C without chord with length at least 5.
In order to make statements simple, we make the following.
− n for any odd cycle C without chord and length at least 5}.
Next we show the following.
Next, let K be an arbitrary maximal clique of G. Set We will show that µ 2 ∈ U (1) . First µ 2 (z) ≥ 1 for any z ∈ V by the definition of µ 2 . Moreover, let K ′ be an arbitrary maximal clique of G.
Further, let C be an arbitrary odd cycle without chord and length at least 5.

Theorem 3.8 E K [HSTAB(G)] is Gorenstein if and only if
(1) Sizes of maximal cliques are constant (say n) and (2) (a) n = 1, (b) n = 2 and there is no odd cycle without chord and length at least 7 or (c) n ≥ 3 and there is no odd cycle without chord and length at least 5.
In particular, if G is an h-perfect graph, then E K [STAB(G)] is Gorenstein if and only if (1) and (2) above are satisfied.
Therefore, #C = 5 and n = 2, since n is an integer. Thus, we see that if n ≥ 3, then there is no odd cycle without chord and length at least 5. Moreover, we see that if n = 2, then there is no odd cycle without chord and length at least 7.
for any e ∈ E and f + (C) ≤ #C−1 2 for any odd cycle C    and QSTAB(G) := f ∈ R V f (x) ≥ 0 for any x ∈ V and f + (K) ≤ 1 for any clique K in G .
If STAB(G) = TSTAB(G), then G is called a t-perfect graph. It is easily verified that HSTAB(G) ⊂ TSTAB(G). In particular, t-perfect graphs are h-perfect. Further, by [Chv,Theorem 3.1], G is perfect if and only if STAB(G) = QSTAB(G). Since HSTAB(G) ⊂ QSTAB(G) by the definition, perfect graphs are h-perfect. (b) i. n = 1, ii. n = 2 and there is no odd cycle without chord and length at least 7 or iii. n = 3 and there is no odd cycle without chord and length at least 5.
(2) (Ohsugi-Hibi) Suppose that G is perfect. Then E K [STAB(G)] is Gorenstein if and only if sizes of maximal cliques are constant.
Proof (1): Since a t-perfect graph is h-perfect and has no clique with size more than 3, the result follows from Theorem 3.8.
(2): Since a perfect graph is h-perfect and has no odd cycle without chord and length at least 5, the result follows from Theorem 3.8.
Remark 3.10 Set K := {K ⊂ V | K is a clique of G and size of K is less than or equal to 3}. Then By defining tU (n) := {µ ∈ Z V − | µ(z) ≥ n for any z ∈ V , µ + (K) ≤ µ(−∞)−n for any maximal element K of K and µ + (C) ≤ µ(−∞) #C−1 2 − n for any odd cycle C without chord and length at least 5} for n ∈ Z, one can verify that T µ ∈ ω (1) Sizes of maximal elements of K are constant (say n) and (2) (a) n = 1, (b) n = 2 and there is no odd cycle without chord and length at least 7 or (c) n = 3 and there is no odd cycle without chord and length at least 5. i.e., (1) E = ∅, (2) G has no isolated vertex nor triangle and there is no odd cycle without chord and length at least 7 or (3) all maximal cliques of G have size at least 3 and there is no odd cycle without chord and length at least 5.