Combinatorial properties of the G-degree

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension $d\ge 4$, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of $(d-1)!$. As a consequence, in even dimension, the terms of the $1/N$ expansion corresponding to odd powers of $1/N$ are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of"associated"cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.


Introduction
It is well-known that regular edge-colored graphs may encode PL-pseudomanifolds, giving rise to a combinatorial representation theory (crystallization theory) for singular PL-manifolds of arbitrary dimension (see Section 2).
In the last decade, the strong interaction between the topology of edge-colored graphs and random tensor models has been deeply investigated, bringing insights in both research fields.
The colored tensor models theory arises as a possible approach to the study of Quantum Gravity: in some sense, its aim is to generalize to higher dimension the matrix models theory which, in dimension two, has shown to be quite useful at providing a framework for Quantum Gravity. The key generalization is the recovery of the so called 1/N expansion in the tensor models context. In matrix models, the 1/N expansion is driven by the genera of the surfaces represented by Feynman graphs; in the higher dimensional setting of tensor models the 1/N expansion is driven by the G-degree of these graphs (see Definition 3), that equals the genus of the represented surface in dimension two.
If (C N ) ⊗d denotes the d-tensor product of the N -dimensional complex space C N , a (d + 1)dimensional colored tensor model is a formal partition function where T belongs to (C N ) ⊗d , T to its dual and B(T, T ) are trace invariants obtained by contracting the indices of the components of T and T . In this framework, colored graphs naturally arise as Feynman graphs encoding tensor trace invariants. As shown in [4], the free energy 1 N d log Z[N, {t B }] is the formal series where the coefficients F ω G [{t B }] are generating functions of connected bipartite (d + 1)-colored graphs with fixed G-degree ω G . The 1/N expansion of formula 1 describes the rôle of colored graphs (and of their G-degree ω G ) within colored tensor models theory and explains the importance of trying to understand which are the manifolds and pseudomanifolds represented by (d + 1)-colored graphs with a given G-degree.
A more detailed description of these relationships between Quantum Gravity via tensor models and topology of colored graphs may be found in [4], [15], [14], [7].
A parallel tensor models theory, involving real tensor variables T ∈ (R N ) ⊗d , has been developed, taking into account also non-bipartite colored graphs (see [17]): this is why both bipartite and nonbipartite colored graphs will be considered within the paper. Section 2 contains a quick review of crystallization theory, including the idea of regular embeddings of edge-colored graphs into surfaces, which is crucial for the definitions of G-degree and regular genus of graphs (Definition 3).
In Section 3, combinatorial properties concerning Hamiltonian decompositions of the complete graph allow to prove the main results of the paper.

Theorem 1 For each bipartite
Note that the above results turn out to have specific importance in the tensor models framework. In fact Theorem 1 implies that, in the d-dimensional complex context, with d even and d ≥ 4, the only non-null terms in the 1/N expansion of formula (1) are the ones corresponding to even (integer) powers of 1/N.
On the other hand, Theorem 2 ensures that in the real tensor models framework, where also nonbipartite graphs are involved, the 1/N expansion contains colored graphs representing (orientable or non-orientable) singular manifolds only in the terms corresponding to even (integer) powers of 1/N. Both Theorems extend to arbitrary even dimension a result proved in [7, Corollary 23] for graphs representing singular 4-manifolds. Section 4 is devoted to the 4-dimensional case: in this particular situation, the general results of Section 3 allow to obtain interesting properties relating the G-degree with the topology of the associated PL 4-manifolds. In fact, the G-degree of a 5-colored graph is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations (Proposition 10). This fact yields relations between these two genera and the Euler characteristic of the associated PL 4-manifold (Proposition 13 and Proposition 14); moreover, two interesting classes of crystallizations arise in a natural way, whose intersection consists in the known class of semi-simple crystallizations, introduced in [2] (see Remark 7 ).

Edge-colored graphs and G-degree
A singular d-manifold is a closed connected d-dimensional polyhedron admitting a simplicial triangulation where the links of vertices are closed connected (d − 1)-manifolds. As a consequence, for each h > 0, the link of any h-simplex is a PL (d − h − 1)-sphere. A vertex whose link is not a PL (d − 1)-sphere is called singular.
Remark 1 If N is a singular d-manifold, then a compact PL d-manifoldŇ is obtained by deleting small open neighbourhoods of its singular vertices. Obviously, N =Ň if and only if N is a closed manifold; otherwise,Ň has a non-empty boundary without spherical components. Conversely, given a compact PL d-manifold M , a singular d-manifold M can be obtained by capping off each component of ∂M by a cone over it.
Note that, in virtue of the above correspondence, a bijection is defined between singular d-manifolds and compact PL d-manifolds with no spherical boundary components. Given a (d + 1)-colored graph (Γ, γ), a d-dimensional pseudocomplex K(Γ) can be associated by the following rules: • for each vertex of Γ, let us consider a d-simplex and label its vertices by the elements of ∆ d ; • for each pair of c-adjacent vertices of Γ (c ∈ ∆ d ), let us glue the corresponding d-simplices along their (d − 1)-dimensional faces opposite to the c-labeled vertices, so that equally labeled vertices are identified.
|K(Γ)| turns out to be a d-pseudomanifold and (Γ, γ) is said to represent it.
Note that, by construction, K(Γ) is endowed with a vertex-labeling by ∆ d that is injective on any simplex. Moreover, a bijective correspondence exists between the h-residues of Γ colored by any B ⊆ ∆ d and the (d − h)-simplices of K(Γ) whose vertices are labeled by ∆ d − B.
In particular, for any color c ∈ ∆ d each connected component of Γĉ is a d-colored graph representing a pseudocomplex that is PL-isomorphic to the link of a c-labeled vertex of K(Γ) in its first barycentric subdivision. Therefore, |K(Γ)| is a singular d-manifold (resp. a closed d-manifold) iff for each color c ∈ ∆ d , allĉ-residues of Γ represent closed (d − 1)-manifolds (resp. the (d − 1)-sphere).
In virtue of the bijection described in Remark 1, a (d + 1)-colored graph (Γ, γ) is said to represent a compact PL d-manifold M with no spherical boundary components if and only if it represents the associated singular manifold M .
The following theorem extends to singular manifolds a well-known result -due to Pezzana ([16])founding the combinatorial representation theory for closed PL-manifolds of arbitrary dimension via colored graphs (the so called crystallization theory).
In particular, each closed PL d-manifold admits a crystallization.
It is well known the existence of a particular set of embeddings of a bipartite (resp. non-bipartite) (d + 1)-colored graph into orientable (resp. non orientable) surfaces.
The Gurau degree (often called degree in the tensor models literature) and the regular genus of a colored graph are defined in terms of the embeddings of Theorem 4.
is called the regular genus of Γ with respect to the permutation ε (i) . Then, the Gurau degree (or G-degree for short) of Γ, denoted by ω G (Γ), is defined as and the regular genus of Γ, denoted by ρ(Γ), is defined as Note that, in dimension 2, any bipartite (resp. non-bipartite) 3-colored graph (Γ, γ) represents an orientable (resp. non-orientable) surface |K(Γ)| and ρ(Γ) = ω G (Γ) is exactly the genus (resp. half the genus) of |K(Γ)|. On the other hand, for d ≥ 3, the G-degree of any (d + 1)-colored graph (resp. the regular genus of any (d + 1)-colored graph representing a closed PL d-manifold) is proved to be a non-negative integer, both in the bipartite and non-bipartite case: see [7, Proposition 7] (resp. [10, Proposition A]).

Proof of the general results
Within combinatorics, the problem of the existence of m-cycle decompositions of the complete graph K n , or of the complete multigraph λK n (i.e. the multigraph with n vertices and with λ edges joining each pair of distinct vertices) is long standing: a survey result, for general m, n and λ, is given in [6, Theorem 1.1].
Moreover, the following results hold, concerning Hamiltonian cycles (i.e. m = n) in K n , both in the case n odd and in the case n even. On the other hand, the following statement regarding the G-degree has been recently proved.
As a consequence, the G-degree of any (d+1)-colored graph is a non-negative integer multiple of (d−1)! 2 .
The result of Proposition 7, which was originally stated in the bipartite case (see [4]), suggested the definition, for d ≥ 3, of the (integer) reduced G-degree which is used by many authors within tensor models theory (see for example [14]). 1 Actually, we are able to prove that, if d ≥ 4 is even, under rather weak hypothesis, the G-degree is multiple of (d-1)! (or, equivalently, the reduced G-degree is even).

Proof of Theorems 1 and 2.
Both in the case of (Γ, γ) bipartite and in the case of (Γ, γ) representing a singular d-manifold, the residues of Γ obviously satisfy the hypotheses of Proposition 8. Hence, if d ≥ 4 is even, ω G (Γ) ≡ 0 mod (d − 1)! holds. ✷ Another particular situation is covered by Proposition 8, as the following corollary explains.
Corollary 9 Let (Γ, γ) be a (d + 1)-colored graph, with d ≥ 4, d even. If (Γ, γ) is a non-bipartite (d + 1)-colored graph such that each d-residue is bipartite, then In the case d ≥ 3 odd, Proposition 6 implies that all d! 2 cyclic permutations (up to inverse) of ∆ d can be partitioned in (d−1)! 2 classes, each containing d cyclic permutations,ε (1) ,ε (2) , . . . ,ε (d) say, so that Hence, a reasoning similar to the one used to prove Proposition 8 yields an alternative proof -for d ≥ 3 odd -of relation (2): since hold, then Remark 3 It is worthwhile to stress that, for d even (resp. odd), formula (3) of Proposition 8 (resp. formula (4) of Remark 2) proves that the sum ) of all regular genera with respect to the d/2 (resp. d) permutations belonging to the same class is half the constant (resp. is the constant) which does not depend on the chosen partition class. Hence, the regular genus ρ(Γ) of the graph Γ is realized by the (not necessarily unique) permutation ε which maximizes the difference where ρε(Γ) denotes the sum of the genera with respect to all other permutations of the same partition class.
Note that, when d = 4, the only partition of all 12 cyclic permutations of ∆ 4 (up to inverse) is given by the 6 classes containing a given permutation ε and its associated ε ′ .
Then, the following result holds.
Proposition 10 For each 5-colored graph (Γ, γ), and for each pair (ε, ε ′ ) of associated cyclic permu- Proof. Equality (5) directly yields As a consequence, the sum of all regular genera of Γ with respect to the 12 cyclic permutations (up to inverse) of ∆ 4 is six times the sum between the regular genera of Γ with respect to any pair ε, ε ′ of associated permutations: ✷ Remark 4 By Proposition 10, for any 5-colored graph the sum between the regular genera of Γ with respect to any pair ε, ε ′ of associated cyclic permutations is a constant (see equality (3) and Remark 3, for d = 4): Hence, the regular genus ρ(Γ) of the graph Γ is realized by the (not necessarily unique) permutation ε so that ρ ε ′ (Γ) − ρ ε (Γ) is maximal.

Moreover:
Proposition 11 (a) If (Γ, γ) is a 5-colored graph, then for each pair (ε, ε ′ ) of associated cyclic permutations of ∆ 4 , γ) is a 5-colored graph representing a singular 4-manifold M 4 , then for each pair (ε, ε ′ ) of associated cyclic permutations of ∆ 4 , Proof. Statement (a) is an easy consequence of Theorem 4: On the other hand, relation 2g r,s,t = g r,s + g r,t + g s,t − p is known to be true for each order 2p 5-colored graph representing a singular 4-manifold (see [7,Lemma 21]. As a consequence we have: By making the difference, is obtained; so, statement (b) follows, via statement (a). ✷ Proposition 11 enables to obtain the following improvement of [7, Proposition 29(a)].
Proof. It is sufficient to apply Proposition 10 to the third equality of [7,Proposition 22]. ✷ Let us now recall two particular types of crystallizations introduced and studied in [2] and [1] 3 : they are proved to be "minimal" with respect to regular genus, among all graphs representing the same PL 4-manifold.
In [2], the relation is proved to hold; hence, p =p + q follows, where q = j,k,l∈∆ 4 t j,k,l ≥ 0 andp = 3χ(M 4 ) + 5(2m − 1) is the minimum possible half order of a crystallization of M 4 , which is attained if and only if M 4 admits semi-simple crystallizations.
With the above notations, the following results can be obtained.
In fact, if q = j,k,l∈∆ 4 t j,k,l ≤ 2, at most two triads (j, k, l) of distinct elements in ∆ 4 exist, so that g j,k,l = 1 + m + t j,k,l > 1 + m. This ensures the existence of a cyclic permutation ε of ∆ 4 so that, for each i ∈ Z 5 , g ε i ,ε i+1 ,ε i+2 = m + 1, which is exactly the requirement for a weak semi-simple crystallization.

Remark 8
The formula obtained in [15,Lemma 4.2] for bipartite (d+1)-colored graphs and extended to the general case in [7,Lemma 13] gives, if d = 4, where, for each i ∈ ∆ 4 , ω G (Γî) denotes the sum of the G-degrees of the connected components of Γî. Hence, i∈∆ 4 ω G (Γî) is always a multiple of 3 (recall Proposition 10 and Theorem 1).