Toric Rings of Perfectly Matchable Subgraph Polytopes

The perfectly matchable subgraph polytope of a graph is a (0,1)-polytope associated with the vertex sets of matchings in the graph. In this paper, we study algebraic properties (compressedness, Gorensteinness) of the toric rings of perfectly matchable subgraph polytopes. In particular, we give a complete characterization of a graph whose perfectly matchable subgraph polytope is compressed.


INTRODUCTION
A lattice polytope P ⊂ R n is a convex polytope such that any vertex of P belongs to Z n .Let K[x ±1 , s] = K[x ±1  1 , . . ., x ±1 n , s] be a Laurent polynomial ring in n + 1 variables over a field K.For a lattice point α = (α 1 , . . . ,α n ) ∈ Z n , we define ±1 , s].If P ∩ Z n = {a 1 , . . ., a m }, then the toric ring K[P] of P is the K-subalgebra of K[x ±1 , s] generated by the monomials x a 1 s, . . ., x a m s ∈ K[x ±1 , s].Furthermore, the toric ideal I P is the defining ideal of K[P], i.e., the kernel of a surjective ring homomorphism π : K[y 1 , . . ., y m ] → K[P] defined by π(y i ) = x a i s for i = 1, 2, . .., m.It is known that I P is generated by homogeneous binomials.See, e.g., [10,30] for details.
Compressed polytopes were defined by Stanley [29] and have been studied from the viewpoint of polyhedral combinatorics, statistics, and optimization.A lattice polytope P is called compressed if the initial ideal of I P is generated by squarefree monomials with respect to any reverse lexicographic order [31].It is known that [30,Corollary 8.9] the initial ideal of I P is generated by squarefree monomials if and only if the corresponding triangulation of P using only the lattice points in P is unimodular.Hence P is compressed if and only if every pulling triangulation of P using only the lattice points in P is unimodular.Sullivant [31] proved that a lattice polytope is compressed if and only if it is 2-level, which is important in optimization theory.For example, the convex polytope of all n × n doubly stochastic matrices, hypersimplices, the order polytopes of finite posets, edge polytopes of bipartite graphs and complete multipartite graphs, and the stable set polytopes of perfect graphs are compressed.
On the other hand, P ⊂ R n is said to be normal if K[P] is a normal semigroup ring.It is known that • P is normal if and only if every vector in kP ∩ L P is a sum of k vectors from P ∩ Z n , where L P is the sublattice of Z n spanned by P ∩ Z n ; • P is normal if there exists a monomial order such that the initial ideal of I P is generated by squarefree monomials.In particular, P is normal if P is compressed.A lattice polytope P ⊂ R n has the integer decomposition property (IDP) if every vector in kP ∩ Z n is a sum of k vectors from P ∩ Z n .In particular, P is normal if P has IDP.However, the converse does not hold in general.
A lattice polytope P ⊂ R n is said to be reflexive if 0 is the unique lattice point in its interior and the dual polytope P * := {x ∈ R n : x • y ≤ 1 for any y ∈ P} is again a lattice polytope.Here x • y is the inner product of x and y.Note that each vertex of P * corresponds to a facet of P. Two lattice polytopes P ⊂ R n and P ′ ⊂ R n ′ are said to be unimodularly equivalent if there exists an affine map from the affine span of P to the affine span aff(P ′ ) of P ′ that maps Z n ∩ aff(P) bijectively onto Z n ′ ∩ aff(P ′ ) and that maps P to P ′ .A lattice polytope P ⊂ R n of dimension n is called Gorenstein of index δ if δ P = {δ a : a ∈ P} is unimodularly equivalent to a reflexive polytope.In particular, a reflexive polytope is Gorenstein of index 1.Note that a lattice polytope P ⊂ R n of dimension n is Gorenstein of index δ if and only if there exist a positive integer δ and a lattice point α ∈ δ (P \ ∂ P) ∩ Z n such that δ P − α is a reflexive polytope, where ∂ P is the boundary of P. Reflexive polytopes are related to mirror symmetry and studied in many areas of mathematics.They are key combinatorial tools for constructing topologically mirror-symmetric pairs of Calabi-Yau varieties, as shown by Batyrev [3].It is known that a lattice polytope P is Gorenstein if and only if the Ehrhart ring , s] of P is Gorenstein.On the other hand, the Ehrhart ring of P coincides with the toric ring of P if and only if P has IDP.
In the present paper, we study conditions for perfectly matchable subgraph polytopes to be compressed or Gorenstein.Let G = (V, E) be a graph on the vertex set V = [n] := {1, 2, . .., n} and the edge set E. Throughout this paper, all graphs are assumed to be finite and simple.A k-matching of G is a set of k pairwise non-adjacent edges of G.If a matching M includes all vertices of G, then M is called a perfect matching.We say that S ⊂ V induces a perfectly matchable subgraph of G if the induced subgraph G[S] of G on the vertex set S has a perfect matching.Let W (G) be the set of all such subsets of V , and adopt the convention that / 0 ∈ W (G), i.e., that the empty subgraph is perfectly matchable.Given a subset S ⊂ V , let ρ(S) = ∑ i∈S e i ∈ R n , where e i is the ith unit vector in R n .In particular, ρ( / 0) is the zero vector.The perfectly matchable subgraph polytope of G, denoted by P G , is the convex hull of {ρ(S) ∈ R n : S ∈ W (G)}.
The perfectly matchable subgraph polytope of a graph is defined in [1].The motivation of their study on perfectly matchable subgraph polytopes is to solve optimization problems that arise in practice (e.g., a bus driver scheduling problem).In optimization theory, compressed polytopes are important since semidefinite programming relaxations are very efficient for compressed polytopes (see, e.g, [8]).
Recently, perfectly matchable subgraphs of graphs appear in the study of h * -polynomials of lattice polytopes.
x |S| , where I G (x) is the interior polynomial of G that is introduced by Kálmán [11] as a version of the Tutte polynomials for hypergraphs.It was shown [12] that the h * -polynomial of the edge polytope of a bipartite graph G coincides with the interior polynomial I G (x) of a hypergraph induced by G. Using these facts, several results on h * -polynomials of several important classes of lattice polytopes are obtained [5,25,26,27].
If G is the disjoint union of graphs G 1 and G 2 , then P G is the product of P G 1 and P G 2 .Hence, P G is compressed (resp.Gorenstein) if and only if both P G 1 and P G 2 are compressed (resp.Gorenstein).Thus, when we are studying such properties, we may assume that G is connected.
The first main result of the present paper is a complete characterization of compressed perfectly matchable subgraph polytopes.A complete k-partite graph denoted by its vertex set such that {s,t} is an edge of G for any s ∈ V i , any t ∈ V j , and any 1 ≤ i < j ≤ k.A complete n-partite graph all of whose independent sets V i have only one vertex is called a complete graph, and denoted by K n .A vertex v of a connected graph G is called a cut vertex if the graph obtained by the removal of v from G is disconnected.Given a graph G, a block of G is a maximal connected subgraph of G without cut vertices.
Theorem 1.1.Let G be a connected graph.Then P G is compressed if and only if all blocks of G are complete bipartite graphs except for at most one block, which is either K 4 or K 1,1,n .
In particular, if P G is compressed, then the line graph of G is perfect by Proposition 4.3.
The second main result of the present paper is a characterization of Gorenstein perfectly matchable subgraph polytopes of bipartite graphs.For any S ⊂ V, let Γ(S) denote the subset of V \ S that consists of vertices adjacent to at least one vertex in S. Theorem 1.2 follows from Proposition 2.2, Theorem 5.4 and Corollary 5.6.Theorem 1.2.Suppose that a connected bipartite graph G = (V 1 ∪ V 2 , E) has a vertex v with deg(v) ≥ 2 such that v is not a cut vertex.Then the following conditions are equivalent: (iv) G has a perfect matching and, for any subset / Moreover, if G is 2-connected, then the above conditions are equivalent to (v) the edge polytope of G is Gorenstein.
The third main result of the present paper is a complete characterization of Gorenstein perfectly matchable subgraph polytopes of pseudotrees.A connected graph which has at most one cycle is called a pseudotree.A regular graph is a graph whose vertices have the same degree.A bidegreed graph is a graph with two different vertex degrees.For example, a path P n and a star graph K 1,n−1 are bidegreed if n ≥ 3.

Theorem 1.3. Let G be a pseudotree on the vertex set V . Then K[P G ] is Gorenstein if and only if G satisfies one of the following:
(i) G is K 1 , K 2 , or a bidegreed tree; (ii) G = C 5 ; (iii) G has a triangle C and (iv) G has an even cycle C, and there exists an integer δ ≥ 2 such that The relationships between toric rings of perfectly matchable subgraph polytopes and toric rings of other polytopes play important roles in this paper.In [1], it was pointed out that P G is unimodulary equivalent to a base polytope of a transversal matroid if G is bipartite.Let G be a graph with the edge set E = {e 1 , . . ., e m }, and let A G = (ρ(e 1 ), . . . ,ρ(e m )) be the matrix associated with the edge polytope Ed(G) of G.It then follows that where Stab(L(G)) is the stable set polytope of the line graph of G (definitions are explained later).From (1.1), it is easy to see that the edge polytope of G is normal if P G is normal (Corollary 2.5).There are many research on the edge polytopes and the stable set polytopes from the point of view of not only discrete geometry but also combinatorial commutative algebra.The study of perfectly matchable subgraph polytopes is expected to contribute the study of these polytopes.The present paper is organized as follows.In Section 2, we introduce relationships between P G and other polytopes.In Section 3, in order to prove the main theorems, we examine inequalities which are facet-inducing for P G .In Section 4, we give a proof for Theorem 1.1.Finally, in Section 5, we give a proof for Theorems 1.2 and 1.3.

RELATIONSHIPS WITH TORIC RINGS OF OTHER POLYTOPES
In this section, we introduce relationships between toric rings of P G and toric rings of other polytopes, i.e., base polytopes of matroids, stable set polytopes, and edge polytopes.
Let G = (V, E) be a graph.Recall that the perfectly matchable subgraph polytope P G of G is the convex hull of {ρ(S) ∈ R n : S ∈ W (G)}, where W (G) is the set of all subsets of V which induce perfectly matchable subsets of G.
Note that compressed (resp.Gorenstein) edge polytopes were studied in [18,21] (resp.[23]).We say that a graph G satisfies the odd cycle condition if, for any two odd cycles where A G = (ρ(e 1 ), . . . ρ(e m )) is the vertex-edge incidence matrix of G. Thus (2.1) In addition, we have Ed(G) ⊂ P G .In such a case, the following holds in general.

Corollary 2.5. Let G be a graph. If P G is normal, then Ed(G) is normal (i.e., G satisfies the odd cycle condition).
Given a lattice polytope P ⊂ R n , let A = (a 1 , . . ., a m ) where P ∩ Z n = {a 1 , . .., a m }.It is known [30] that the toric ideal I P of P is generated by binomials y a −y b ∈ K[y 1 , . . ., y m ] such that A(a − b) = 0 and deg y a = deg y b .
If G is not a tree, then n = m and hence A G is an m ×m matrix.It is known [10, Lemmas 5.5 and 5.6] that A G is a regular matrix if and only if G is a pseudotree which has no even cycles.If G is a tree, then n = m + 1 and A G is an (m + 1) × m matrix of rank m.
Since the rank of the Proposition 2.7.Let G be a pseudotree.Then P G is normal.
Proof.From Proposition 2.2, we may assume that G is not bipartite.Then, by Proposition 2.

FACETS OF PERFECTLY MATCHABLE SUBGRAPH POLYTOPES
In this section, we introduce inequalities for the facets of P G given in [1,2].We will see that these inequalities depend on whether G is bipartite.Let has an odd number of vertices}.For any A ⊂ V, let Γ(A) denote the subset of V \ A that consists of vertices adjacent to at least one vertex in A. For any S ⊂ V , let θ (S) be the number of connected components of the induced subgraph G[S].

for all S ∈ T such that every component of G[S] consists of a single vertex or else is a nonbipartite graph with an odd number of vertices.
A graph G = (V, E) is called critical (or hypomatchable) if, for every v ∈ V , G \ {v} has a perfect matching.A critical graph is a nonbipartite graph with an odd number of vertices.

Proposition 3.2 ([2]
).Let G be a nonbipartite graph.For S ∈ T , the inequality (3.2) is facet-inducing for P G if and only if S satisfies the following conditions: ] by deleting all edges with both ends in Γ(S) is connected.

Proposition 4.1 ([31]
).Let P be a lattice polytope having the irredundant linear description P = {x ∈ R n : a i • x ≥ b i , i = 1, . .., s}, where a i ∈ R n for i = 1, 2, . .., s.In addition, let L ⊂ Z n be a lattice spanned by P ∩ Z n .Then P is compressed if and only if, for each i, there is at most one nonzero m i ∈ R such that {x ∈ L : Let G ′ be an induced subgraph of a graph G. Then P G ′ is a face of P G .It is known that every face of a compressed polytope is compressed.Hence, we have the following immediately.
Lemma 4.2.Let G ′ be a connected graph such that P G ′ is not compressed.If a graph G has G ′ as an induced subgraph, then P G is not compressed.
The following fact is known in graph theory.Proposition 4.3 ([15,32]).Let G be a graph.Then the following conditions are equivalent: (i) is facet-inducing for P G ′ from Proposition 3.2.Then there exist n + 1 ≥ 3 kinds of values for x(S) with x ∈ P G ′ ∩ Z 2n+1 .In fact, we have From Proposition 4.1, P G ′ is not compressed.
Lemma 4.5.Let G be a connected graph.Suppose that P G is compressed.Then for any even cycle C in G of length 2n ≥ 6, the induced subgraph G[V (C)] is a complete bipartite graph K n,n .
Proof.Suppose that P G is compressed.Let C = (v 1 , . . ., v 2n ) be an even cycle in G of length 2n ≥ 6, and let We prove the statement by induction on n.From Lemma 4.2, P G ′ is compressed.Note that G ′ is a subgraph of a block of G. From Proposition 4.3 and Lemma 4.4, G ′ is a bipartite graph since neither K 4 nor K 1,1,n has an even cycle of length ≥ 6. Suppose that G ′ is not a complete bipartite graph.
Case 2. (n ≥ 4 and suppose that the statement is true for any even cycle of length ≤ 2n−2) Suppose that {v 3 , v 2k } for some 3 ≤ k ≤ n is not an edge of G ′ .If {v 3 , v 2k ′ } is an edge of G ′ for some k ′ , then v 3 , v 2k are contained in an even cycle of length 2m with 6 ≤ 2m ≤ 2n − 2. By the hypothesis of induction, {v 3 , v 2k } is an edge of G ′ , a contradiction.Thus, for any 3 Hence, P G ′ is not compressed, a contradiction.Thus, G ′ is a complete bipartite graph K n,n .Lemma 4.6.Let G be a connected graph.If P G is compressed, then any two triangles of G have a common edge.
Proof.Suppose that P G is compressed and two triangles C and C ′ of G have no common edges.

Case 1. (C and C ′ have exactly one common vertex)
Let Then the vertex set and the edge set of G ′ are Since G has no odd cycle of length ≥ 5 as a subgraph, G ′ is an induced subgraph of G, and hence P G ′ is compressed.
We now consider the facets of P G ′ .Let S = {v 3 }.Since |S| = 1, the set S satisfies conditions (i) and (iii) in Lemma 3.2.In addition, since G ′ \ (S ∪Γ(S)) is empty, S satisfies condition (ii) in Lemma 3.2.Thus, S induces a facet of P G ′ .However, we have Hence, P G ′ is not compressed, a contradiction.

Case 2. (C and C ′ have no common vertices)
Since G is connected, there exists a path We may assume that s (≥ 1) is minimal among pairs of triangles without common edges.Let G ′′ be an induced subgraph on the vertex set V (C) ∪V (C ′ ) ∪V (P).Let S = {v 2 }.Since |S| = 1, the set S satisfies conditions (i) and (iii) in Lemma 3.2.
For this case, G ′′ \ (S ∪ Γ(S)) is nonbipartite.However, we have There exists an edge {v 2 , v}, where v ( = v 1 ) belongs to either the path P or . This contradicts the hypothesis that s is minimal.
We are now in a position to prove a main theorem.
Proof of Theorem 1.1.("Only if") Suppose that P G is compressed.From Proposition 4.3 and Lemma 4.4, each block of G is either a bipartite graph, K 4 , or K 1,1,n .By Lemma 4.6, at most one block is either K 4 or K 1,1,n .It is enough to show that each bipartite block is a complete bipartite graph.Let B be a bipartite block of G on the vertex set B 1 ∪ B 2 .Suppose that {i, j} is not an edge of G for vertices i ∈ B 1 and j ∈ B 2 .Since B is 2connected, there exist two disjoint paths P 1 and P 2 from i to j in B. Note that the length of each P i is at least 3. Hence P 1 ∪ P 2 is an even cycle of length ≥ 6.This contradicts to Lemma 4.5.Thus, B is a complete bipartite graph.
("if") Suppose that all blocks of G are complete bipartite graphs except for at most one block, which is either K 4 or K 1,1,n and P G is not compressed.

Case 1. (G is bipartite) There
is facet-inducing, and x(S) − x(Γ(S)) ≤ −2 for some x.It then follows that there exist four distinct vertices i, i ′ ∈ Γ(S) and j, j are connected.Then there exists an even cycle where

Case 2. (G is not bipartite)
There exists a subset S ⊂ V such that   )] has an edge {i, j}.Since the graph obtained from G by deleting all edges with both ends in Γ(S) is connected, there exists a path P from i to j which does not contain {i, j}.Then T 2 = P ∪ {i, j} is an odd cycle of G. Since the length of T 2 is 3, and T 1 and T 2 have no common edge, this is a contradiction.must be even number.Since P G is not compressed, there exists x such that Hence, G[Γ(S)] has two edges without a common vertex and hence G has two triangles without a common edge.This is a contradiction.
Example 4.7.The perfectly matchable subgraph polytope P G of the graph G in Figure 1 is compressed.

GORENSTEIN PERFECTLY MATCHABLE SUBGRAPH POLYTOPES
In this section, for several classes of graphs, we give a characterization of a graph G such that K[P G ] is Gorenstein.If G is either K 1 or K 2 , then K[P G ] is isomorphic to a polynomial ring and hence K[P G ] is Gorenstein.Throughout this section, we may assume that G has at least two edges.5.1.2-connected bipartite graphs.Suppose that G is a bipartite graph on the vertex set V = V 1 ∪V 2 = {1, . . ., n}, where n ∈ V 2 .Then P G lies on the hyperplane H defined by the equation x(V 1 ) = x(V 2 ).Let ψ : R n−1 → H denote the affine map defined by setting If G is bipartite, we have the following criterion for G whose P G is Gorenstein.Note that any vertex of degree one is not a cut vertex.Proposition 5.1.Let G be a connected bipartite graph on the vertex set V = {1, 2, . .., n} = V 1 ∪ V 2 .Then P G is Gorenstein of index δ if and only if δ ≥ 2 and there exists α ∈ Z n such that the following hold: Proof.Let P = ψ −1 (P G ), where ψ is the map defined as above.Then P is Gorenstein of index δ if and only if there exists a lattice point β ∈ δ (P \ ∂ P) ∩ Z n−1 such that δ P − β is a reflexive polytope, where δ P = {δ a : a ∈ P}.
By Proposition 3.3, substituting Then S satisfies one of the following: , where v ∈ Γ(S).Since S and Γ(S) give a partition of the vertex set of bipartite graph G[S ∪ Γ(S)], the sum of the degree sequence of S is equal to that of   ("Only if") Suppose that P G is Gorenstein.Since any even cycle satisfies condition (iv) in Theorem 1.3, we may assume that G is not an even cycle.Then we have V p = / 0. By Proposition 5.1, there exist δ and α satisfying conditions (i)-(iv).It then follows that δ P G has α ∈ Z n , where as an interior lattice point.Example 5.9.The perfectly matchable subgraph polytope P G of the graph G in Figure 3 is Gorenstein.
6, K[P G ] ∼ = K[Stab(L(G))].It is known [6, Theorem 8.1] that the toric ideal of the stable set polytope of an almost bipartite graph has a squarefree quadratic initial ideal.It then follows that Stab(C 2n+1 ) is normal.In addition, Stab(K m ) is a simplex and hence normal.It is known [16, Proposition 1] that the stable set polytope of the clique-sum of simple graphs G 1 and G 2 is normal if and only if both Stab(G 1 ) and Stab(G 2 ) are normal.Since L(G) is a clique-sum of an odd cycle C 2n+1 = L(C 2n+1 ) and some cliques, Stab(L(G)) is normal.
[22] C 2 in the same connected component of G without common vertices, there exists an edge {i, j} of G such that i ∈ V (C 1 ) and j ∈ V (C 2 ).The stable set polytope of G is the convex hull of the set {ρ(S 1 ), ..., ρ(S t )}, denoted by Stab(G).It is known (e.g.,[22]) that Stab(G) is compressed if and only if G is perfect.Moreover, it is known [23, Theorem 1.2 (b)] that, for any perfect graph G, Stab(G) is Gorenstein if and only if all maximal cliques of G have the same cardinality.The line graph L A finite subset S ⊂ V is called stable in G if none of the edges of G is a subset of S. In particular, the empty set / 0 is stable.Let S(G) = {S 1 , . .., S t } denote the set of all stable sets of G.
Each block of G is either a bipartite graph, K 4 , or K 1,1,n .It is known that Stab(G) is compressed if and only if G is perfect.
Lemma 4.4.Let G be a connected graph.If P G is compressed, then L(G) is perfect and hence Stab(L(G)) is compressed.Proof.Suppose that L(G) is not perfect.From Proposition 4.3, G has an odd cycle C 2n+1 with n ≥ 2 as a subgraph.Let S = V (C 2n+1 ) and G ′ = G[S].From Lemma 4.2, it is enough to prove that P G ′ arising from the induced subgraph