Combinatorial properties of the G-degree
Maria Rita Casali Luigi Grasselli
math.GT 57Q15, 57N13, 57M15, 83E99




 
 
 
1] Maria Rita Casali
2] Luigi Grasselli

[1] Department of Physics, Mathematics and Computer Science, University of Modena and Reggio Emilia,          Via Campi 213 B - 41125 Modena (Italy), casali@unimore.it

[2] Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia,                             Via Amendola 2, Pad. Morselli - 42122 Reggio Emilia (Italy), luigi.grasselli@unimore.it


Combinatorial properties of the G-degree
    [
    December 30, 2023
========================================




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 G-degree of the involved graphs, which drives the 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≥ 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.
 


  Keywords: edge-colored graph; PL-manifold; singular manifold; colored tensor model; 
  regular genus; Gurau-degree. 

2000 Mathematics Subject Classification:
   57Q15 - 57N13 - 57M15 - 83E99.



§ 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 <ref>).


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 <ref>), 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

    𝒵[N,{α_B}]:=∫_dTdT/(2π)^N^dexp(-N^d-1T· T + ∑_Bα_BB(T,T)),

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 <cit.>, the free energy  1/N^dlog𝒵[N,{t_B}] is the formal series

    1/N^dlog𝒵[N,{t_B}] = ∑_ω_G≥ 0N^-2/(d-1)!ω_GF_ω_G[{t_B}]∈ℂ[[N^-1, {t_B}]],

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 (<ref>) 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 <cit.>, <cit.>, <cit.>, <cit.>.

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 <cit.>): this is why both bipartite and non-bipartite colored graphs will be considered within the paper.



Section <ref> contains a quick review of crystallization theory, including the idea  of regular embedding of edge-colored graphs into surfaces, which is crucial for the definitions of G-degree and regular genus of graphs (Definition <ref>).

In Section <ref>, combinatorial properties concerning Hamiltonian decompositions of the complete graph allow to prove the main results of the paper.



 
For each  bipartite (d+1)-colored graph (Γ,γ),  with d even, d ≥ 4,  
    ω_G (Γ) ≡ 0  (d-1)!



 
For each  (bipartite or non-bipartite) (d+1)-colored graph (Γ,γ) representing a singular d-manifold, with d even, d≥ 4,   
    ω_G (Γ) ≡ 0  (d-1)!




Note that the above results turn out to have specific importance in the tensor models framework. In fact Theorem <ref> 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 (<ref>) are the ones corresponding to even (integer) powers of 1/N.

On the other hand, Theorem <ref> ensures that in the real tensor models framework, where also non-bipartite graphs are involved, the 1/N expansion contains colored graphs representing (orientable or non-orientable)  singular manifolds - and, in particular, closed manifolds - only in the terms corresponding to even (integer) powers of 1/N.
Both Theorems extend to arbitrary even dimension a result proved in <cit.> for graphs representing singular 4-manifolds.



Section <ref> is devoted to the 4-dimensional case: in this particular situation, the general results of  Section <ref> 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 <ref>). This fact yields relations between these two genera and the Euler characteristic of  the associated PL 4-manifold (Proposition <ref> and Proposition <ref>); moreover, two interesting classes of crystallizations arise in a natural way,  whose intersection consists in the known class of semi-simple crystallizations, introduced in <cit.> (see Remark <ref>).





§ EDGE-COLORED GRAPHS AND G-DEGREE



A singular d-manifold is a compact connected d-dimensional polyhedron admitting a simplicial triangulation where the links of vertices are closed connected (d-1)-manifolds, while the link of any h-simplex, for each h > 0, is a PL (d-h-1)-sphere.  A vertex whose link is not a PL (d-1)-sphere is called singular.


 
The class of singular d-manifolds includes the class of closed d-manifolds: in fact, a closed d-manifold is a singular d-manifold without singular vertices. 
Moreover, if N is a singular d-manifold, then a compact PL d-manifold Ň 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.




 
A (d+1)-colored graph (d ≥ 2) is a pair (Γ,γ), where Γ=(V(Γ), E(Γ)) is a regular d+1 valent multigraph  (i.e. multiple edges are allowed, while loops are forbidden)  and γ: E(Γ) →Δ_d={0,…, d} is a map injective on adjacent edges, called coloration.




For every ℬ⊆Δ_d let Γ_ℬ be the subgraph obtained from (Γ, γ) by deleting all the edges colored by Δ_d - ℬ. The connected components of Γ_ℬ are called ℬ-residues or, if #ℬ = h, h-residues of Γ; the symbol g_ℬ denotes their number. In the following, if ℬ ={c_1,…,c_h}, its complementary set Δ_d - ℬ will be denoted by ĉ_1…ĉ_h.



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 Δ_d;

  * for each pair of c-adjacent vertices of Γ (c∈Δ_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[In fact, |K(Γ)| is a quasi-manifold: see  <cit.>.], which  is orientable if and only if Γ is bipartite,  and (Γ, γ) is said to represent it.



Note that, by construction, K(Γ) is endowed with a vertex-labeling by Δ_d that is injective on any simplex.
Moreover, a bijective correspondence exists between the h-residues of Γ colored by any ℬ⊆Δ_d and the (d-h)-simplices of K(Γ) whose vertices are labeled by Δ_d - ℬ.


In particular, for any color c∈Δ_d, each connected component of Γ_ĉ is a d-colored graph representing  a pseudocomplex that is PL-homeomorphic to the link of a c-labeled vertex of K(Γ) in its first barycentric subdivision.
As a consequence, |K(Γ)| is a singular d-manifold (resp. a closed d-manifold) iff for each color
c∈Δ_d, all ĉ-residues of Γ represent closed (d-1)-manifolds (resp. the (d-1)-sphere).


In virtue of the bijection described in Remark <ref>, 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.


 
A crystallization of a closed PL d-manifold M^d is a (d+1)-colored graph representing M^d, such that each d-residue is connected (i.e.  g_î=1  ∀ i ∈Δ_d).



The following theorem extends to singular manifolds a well-known result - due to Pezzana (<cit.>) - founding the combinatorial representation theory for closed PL-manifolds of arbitrary dimension via colored graphs (the so called crystallization theory).  See also <cit.> and <cit.> for the 3-dimensional case.


 (<cit.>) 
Any singular d-manifold  - or, equivalently, any compact d-manifold with no spherical boundary components -  admits a (d+1)-colored graph representing it.

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.

(<cit.>)
Let (Γ,γ) be a bipartite (resp. non-bipartite) (d+1)-colored graph of order 2p. Then for each cyclic permutation ε = (ε_0,…,ε_d) of Δ_d, up to inverse, there exists a cellular embedding, called regular, of (Γ,γ) into an orientable (resp. non-orientable) closed surface F_ε(Γ)
whose regions are bounded by the images of the {ε_j,ε_j+1}-colored cycles, for each j ∈ℤ_d+1.
Moreover, the genus (resp. half the genus)  ρ_ε (Γ) of F_ε(Γ) satisfies


    χ (F_ε(Γ)) =  2 - 2ρ_ε(Γ)= ∑_j∈ℤ_d+1 g_ε_jε_j+1 + (1-d)p.


No regular embeddings of (Γ,γ) exist into non-orientable (resp. 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 <ref>.

 
Let (Γ,γ) be a (d+1)-colored graph.
If {ε^(1), ε^(2), … , ε^(d!/2)} is the set of all cyclic permutations of Δ_d (up to inverse), ρ_ε^(i) (i=1, … , d!/2) 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

    ω_G(Γ)  =  ∑_i=1^d!/2ρ_ε^(i)(Γ)

and the regular genus of Γ, denoted by ρ(Γ), is defined as

    ρ(Γ)  =  min {ρ_ε^(i)(Γ) / i=1,…,d!/2}.




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 <cit.> (resp. <cit.>).





§ 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 λ K_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 <cit.>.

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.

  <cit.> 
For all odd n ≥ 3 there exists a partition of all Hamiltonian cycles
of K_n into (n-2)!  Hamiltonian cycle decompositions of K_n.


  <cit.> 
For all even n ≥ 4 there exists a partition of all Hamiltonian cycles of K_n into (n-2)!/2 classes, so that each edge of K_n appears in exactly two cycles belonging to the same class.



Figure 1 describes - as an example of Proposition <ref>  - the six Hamiltonian cycle decompositions of K_5, each of them containing a pair of disjoint Hamiltonian cycles (given by the dashed and continuous edges respectively).  
Note that, by labelling the vertices of K_n with the elements of Δ_n-1, each Hamiltonian cycle in K_n defines a cyclic permutation of Δ_n-1 (together with its inverse), and viceversa. 





 


On the other hand, the following statement regarding the G-degree has been recently proved.

  <cit.> 
If (Γ, γ) is a (d+1)-colored graph  of order 2p (d≥ 3), then

    ω_G(Γ)  =  (d-1)!/2·(d + p · (d-1) ·d/2  -  ∑_r,s ∈Δ_d g_rs).

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 <ref>, which was originally stated in the bipartite case (see <cit.>), suggested the definition, for d≥ 3, of the (integer) reduced G-degree

    ω^'_G(Γ)  =  2/(d-1)!·ω_G(Γ),

which is used by many authors within tensor models theory (see for example <cit.>).[In fact, the exponents of N^-1 in the  1/N expansion of formula (<ref>) are all (non-negative) integers ω_G^'.]


Actually, we are able to prove that, if d ≥ 4 is even, under 
rather weak hypotheses, 
the G-degree is multiple of (d-1)! (or, equivalently, the reduced G-degree is even).


 
If d ≥ 4 is even, and  (Γ, γ) is a (d+1)-colored graph such that 
each 3-residue is bipartite and each d-residue is either bipartite or non-bipartite with integer regular genus with respect to any permutation, 
then

    ω_G(Γ) ≡ 0  (d-1)!




Let (Γ,γ) be a (d+1)-colored graph, with d ≥ 4, d even.
Since n = d+1 is odd, Proposition <ref> implies that all d!/2 cyclic permutations (up to inverse) of Δ_d can be partitioned in (d-1)! classes, each containing d/2 cyclic permutations,
ε̅^(1), ε̅^(2), … , ε̅^(d/2) say,
so that

    ∑_i=1^d/2[ ∑_j∈ℤ_d+1 g_ε̅^(i)_j,ε̅^(i)_j+1] = ∑_r,s∈Δ_d g_r,s.

Hence, by Theorem <ref>,

    2 ·d/2 - 2 ·∑_i=1^d/2ρ_ε̅^(i) =  ∑_r,s ∈Δ_d g_rs   + p · (1-d) ·d/2,

and

    2 ·∑_i=1^d/2ρ_ε̅^(i) =  d + p · (d-1) ·d/2  -  ∑_r,s ∈Δ_d g_rs.

This proves that the quantity  ∑_i=1^d/2ρ_ε̅^(i) is constant, for each class of the above partition; then,

    ω_G (Γ)=  (d-1)! ·∑_i=1^d/2ρ_ε̅^(i)

immediately follows.[In this way, one can reobtain - for d ≥ 4 even - relation (<ref>).] 

Now, the hypotheses on the 3-residues and d-residues 
of Γ directly implies, in virtue of  Theorem <ref> and  of <cit.> (originally proved in <cit.>),
that  ∑_i=1^d/2ρ_ε̅^(i) is an integer, and hence ω_G (Γ) ≡ 0  (d-1)!   (or equivalently, ω_G^' (Γ) = 2 ·ω_G (Γ)/(d-1)! is even).





Then, the results stated in Section <ref> trivially follow. 



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 <ref>.   
Hence, if d≥ 4 is even,  ω_G(Γ) ≡ 0  (d-1)! holds. 



Another particular situation is covered by Proposition  <ref>, as the following corollary explains.


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 
    ω_G (Γ) ≡ 0  (d-1)!

-0.3truecm   


Note that there exist (d+1)-colored graphs, with d even, d ≥ 4 and odd reduced G-degree: of course, in virtue of Theorems 1 and 2, the represented d-pseudomanifold must be non-orientable and it can't be a singular d-manifold. As an example, for each d ≥ 4, the (d+1)-colored graph  (Γ,γ) of Figure 2 represents the (d-2)-th suspension Σ of the real projective plane ℝℙ^2  and ω_G^'(Γ)=d-1. 
It is easy to check that Γ is non-bipartite and Σ is not a singular manifold, since all d-residues of Γ, with the exception of Γ_0̂, do not represent a closed (d-1)-manifold. 
Moreover, its (unique) {0,1,c }-residue, for any c ∈{2, …, d}, is non-bipartite and represents  the  non-orientable genus one surface ℝℙ^2. 








  In the case d ≥ 3 odd, Proposition <ref> implies that all d!/2 cyclic permutations (up to inverse) of Δ_d can be partitioned in (d-1)!/2 classes, each containing d cyclic permutations,
ε̅^(1), ε̅^(2), … , ε̅^(d) say,
so that

    ∑_i=1^d[ ∑_j∈ℤ_d+1 g_ε̅^(i)_j,ε̅^(i)_j+1] = 2 ∑_r,s∈Δ_d g_r,s.

Hence, a reasoning similar to the one used to prove Proposition <ref> yields an alternative proof - for d ≥ 3 odd - of relation (<ref>): 
since 
    2 · d - 2 ·∑_i=1^d ρ_ε̅^(i) = 2 ∑_r,s ∈Δ_d g_rs   + p · (1-d) · d

and

    ∑_i=1^d ρ_ε̅^(i)  =  d + p · (d-1) ·d/2  -  ∑_r,s ∈Δ_d g_rs

hold, then

    ω_G (Γ)= (d-1)!/2·∑_i=1^d ρ_ε̅^(i) = (d-1)!/2·(d + p · (d-1) ·d/2 -  ∑_r,s ∈Δ_d g_rs)

directly follows.


  It is worthwhile to stress that, for d even (resp. odd), formula (<ref>) of Proposition <ref> (resp. formula (<ref>) of Remark <ref>) proves that the sum ∑_i=1^d/2ρ_ε̅^(i) (resp. ∑_i=1^d ρ_ε̅^(i)) of all regular genera with respect to the d/2 (resp. d) permutations belonging to the same class
is half the reduced G-degree (resp. is the reduced G-degree)

    ω^'_G(Γ)  =  d + p · (d-1) ·d/2  -  ∑_r,s ∈Δ_d g_rs,
  
i.e. it is a 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

    ρ_ε̂(Γ) - ρ_ε(Γ),

where ρ_ε̂(Γ) denotes the sum of the genera with respect to all other permutations of the same partition class.





§ THE 4-DIMENSIONAL CASE
  


In the 4-dimensional setting, the above combinatorial properties allow to prove further results about the G-degree.

In fact, it is easy to check that, for each  cyclic permutation ε = (ε_0, ε_1, ε_2, ε_3, ε_4) of Δ_4, we have

    { (ε_j,ε_j+1)  /  j∈ℤ_5} ∪ { (ε_j, ε_j+2)  / j∈ℤ_5} =  { (i,j)  /   i,j∈Δ_5, i  j}.


Let us denote by ε^'=(ε^'_0, ε^'_1, ε^'_2, ε^'_3, ε^'_4)  the permutation (ε_0, ε_2, ε_4, ε_1, ε_3) of Δ_4, which may be said to be associated to ε.

Note that, when d=4, the only partition of all 12 cyclic permutations of Δ_4 (up to inverse) is given by the six classes containing a given permutation ε and its associated ε^': see Figure 1, where each class of the Hamiltonian cycle decomposition of K_5 ensured by Proposition <ref>  is shown to correspond to a pair (ε_i, ε_i^') of associated permutations, for i ∈ℕ_6. For example, the first class corresponds to the identical permutation ε_1 = (0,1,2,3,4) and its associated permutation ε_1^' = (0,2,4,1,3). 



Then, the following result holds.

 
For each 5-colored graph  (Γ, γ),  and for each pair (ε, ε^') of associated cyclic permutations of Δ_4,

    ω_G(Γ)   =  6 (ρ_ε(Γ) + ρ_ε^'(Γ)).

 



Equality (<ref>) directly yields

    ∑_j∈ℤ_5 g_ε_j,ε_j+1 +  ∑_j∈ℤ_5 g_ε^'_j,ε^'_j+1= ∑_i,j∈Δ_5 g_i,j.


As a consequence, the sum of all regular genera of Γ with respect to the 12 cyclic permutations (up to inverse) of Δ_4  is six times the sum between the regular genera of Γ with respect to any pair ε, ε^' of associated permutations:

    ω_G(Γ)   =  6 (ρ_ε(Γ) + ρ_ε^'(Γ)).

-0.6truecm   


  By Proposition <ref>, 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
constant (see equality (<ref>) and Remark <ref>, for d=4):

    ρ_ε(Γ) + ρ_ε^'(Γ)  =  1/2ω'_G(Γ) =  2+3p - 1/2∑_i,j∈Δ_5 g_i,j.

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:
 


  (a)
If  (Γ, γ) is a 5-colored graph, then for each pair (ε, ε^') of associated cyclic permutations of Δ_4,

    2 (ρ_ε^'(Γ) - ρ_ε(Γ) )   =  ∑_j∈ℤ_5 g_ε_j,ε_j+1 - ∑_j∈ℤ_5 g_ε_j,ε_j+2


  (b)
If  (Γ, γ) is a 5-colored graph representing a singular 4-manifold M^4,  then for each pair (ε, ε^') of associated cyclic permutations of Δ_4,

    ρ_ε^'(Γ) - ρ_ε(Γ)   =  ∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2 - ∑_j∈ℤ_5 g_ε_j,ε_j+2, ε_j+4






Statement (a) is an easy consequence of Theorem <ref>:

    2-2 ρ_ε(Γ) = ∑_j∈ℤ_5 g_ε_j,ε_j+1 -3p

and

    2-2 ρ_ε^'(Γ) = ∑_j∈ℤ_5 g_ε^'_j,ε^'_j+1 -3p=  ∑_j∈ℤ_5 g_ε_j,ε_j+2 -3p.
   

On the other hand, relation 2  g_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 <cit.>. As a consequence we have:

    2 ∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2= ∑_r,s∈ℤ_5 g_r,s + ∑_j∈ℤ_5 g_ε_j,ε_j+1 - 5p
  and

    2 ∑_j∈ℤ_5 g_ε_j,ε_j+2,ε_j+4= ∑_r,s∈ℤ_5 g_r,s + ∑_j∈ℤ_5 g_ε_j,ε_j+2 - 5p.

By making the difference,

    2 (∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2 - ∑_j∈ℤ_5 g_ε_j,ε_j+2,ε_j+4) =  ∑_j∈ℤ_5 g_ε_j,ε_j+1 - ∑_j∈ℤ_5 g_ε_j,ε_j+2
 is obtained; so, statement (b) follows, via statement (a).


Proposition <ref> enables to obtain
the following improvement of  <cit.>.


Let (Γ,γ) be a 5-colored graph.  Then:

    ω_G(Γ)  =  12 ·ρ (Γ)    ⟺


    ∑_j∈ℤ_5 g_ε_j, ε_j+1 =  ∑_j∈ℤ_5 g_ε_j, ε_j+2 for each cyclic permutation ε of Δ_4.




From <cit.>, it is known that

    ω_G(Γ)  =  12 ·ρ (Γ)    ⟹    ∑_j ∈ℤ_5 g_ε̅_j, ε̅_j+1 =  ∑_j∈ℤ_5 g_ε̅_j, ε̅_j+2,

ε̅ being the cyclic permutation of Δ_4 such that ρ(Γ) = ρ_ε̅(Γ).
So, ρ_ε̅^'(Γ) - ρ_ε̅(Γ) = 0 directly follows via Proposition <ref> (a).
Now, since ρ_ε(Γ) + ρ_ε^'(Γ) is constant for each pair (ε, ε^') of associated cyclic permutations of Δ_4 (see Remark <ref>), then:

    |ρ_ε^'(Γ) - ρ_ε(Γ)| ≤ρ_ε̅^'(Γ) - ρ_ε̅(Γ).

Hence, ρ_ε̅^'(Γ) - ρ_ε̅(Γ) = 0 implies ∑_j∈ℤ_5 g_ε_j, ε_j+1 =  ∑_j∈ℤ_5 g_ε_j, ε_j+2 (i.e. ρ_ε^'(Γ) - ρ_ε(Γ) = 0, in virtue of Proposition <ref> (a)) for each cyclic permutation ε of Δ_4.

The reversed implication is straightforward, via Proposition <ref>(a).





 
If (Γ, γ)  is an order 2p 5-colored graph representing
a singular 4-manifold M^4, then, for each pair (ε, ε^') of associated cyclic permutations of Δ_4:

    χ(M^4)   =  (ρ_ε(Γ) + ρ_ε^'(Γ)) -p + ∑_i∈Δ_4  g_î -2.

 


It is sufficient to apply  Proposition <ref>  to the third equality of <cit.>.




Let us now recall two particular types of crystallizations introduced and studied in <cit.> and <cit.>[Both semi-simple and weak semi-simple crystallizations generalize the notion of simple crystallizations for simply-connected PL 4-manifolds: see <cit.> and <cit.>.]: they are proved to be “minimal" with respect to regular genus, among all graphs representing the same PL 4-manifold.

 A crystallization of a PL 4-manifold M^4 with   rk(π_1(M^4))= m ≥ 0 is called a semi-simple crystallization
if  g_j,k,l = 1 + m   ∀ j,k,l ∈Δ_4.

A crystallization of a PL 4-manifold M^4 with   rk(π_1(M^4))= m is called a weak semi-simple crystallization
if  g_i, i+1, i +2 = 1 + m   ∀ i ∈Δ_4 (where the additions in subscripts are intended in ℤ_5).



According to <cit.>, for each order 2p crystallization  (Γ, γ) of a closed  PL 4-manifold M^4, with   rk(π_1(M^4))= m0, let us set

    g_j,k,l= 1+m + t_j,k,l,    with t_j,k,l  ∀ j,k,l ∈Δ_4.

Semi-simple (resp. weak semi-simple) crystallizations turn out to be characterized by t_j,k,l = 0   ∀ j,k,l ∈Δ_4 (resp. t_i, i+1, i +2 = 0  ∀ i ∈ℤ_5).

In <cit.>, the relation

    p= 3 χ(M^4)+ 5(2m -1) +  ∑_j,k,l ∈Δ_4 t_j,k,l

is proved to hold; hence, p = p̅ + q follows, where q= ∑_j,k,l ∈Δ_4 t_j,k,l≥ 0 and p̅= 3 χ(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.

 
Let (Γ,γ) be an order 2p crystallization  of a closed PL 4-manifold M^4, with rk(π_1(M^4))=m. Then, for each pair (ε, ε^') of associated cyclic permutations of Δ_4, with ρ_ε(Γ) ≤ρ_ε^'(Γ):

    ρ_ε^'(Γ) - ρ_ε(Γ) = q - 2 ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4 ≤ q.

Moreover, The following statements are equivalent:


  (a) a cyclic permutation ε of Δ_4 exists, such that ρ_ε^'(Γ) - ρ_ε(Γ) =q;

  (b)  Γ is a weak semi-simple crystallization;

  (c) ρ(Γ)= 2χ(M^4) + 5 m -4.[Note that, in this case, the permutation ε̅ such that ρ(Γ)= ρ_ε̅(Γ) coincides with the permutation ε of point (a); moreover, ρ(Γ)= 𝒢(M^4) holds, in virtue of the inequality  𝒢(M^4) ≥ 2χ(M^4) + 5 m -4, proved in  <cit.>.]




In virtue of Proposition <ref>(b), an easy computation proves relation (<ref>):

    ρ_ε^'(Γ) - ρ_ε(Γ)      =  ∑_i∈ℤ_5  (m+1 + t_ε_i,ε_i+1,ε_i+2 ) - ∑_i∈ℤ_5 (m+1 + t_ε_i,ε_i+2,ε_i+4)  = 
         =  ∑_i∈ℤ_5  t_ε_i,ε_i+1,ε_i+2 - ∑_i∈ℤ_5   t_ε_i,ε_i+2,ε_i+4 = 
         =  ∑_j,k,l ∈Δ_4  t_j,k,l - 2 ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4 = 
         =   q - 2 ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4 ≤  q.


Now, if Γ is a weak semi-simple crystallization, by definition itself a cyclic permutation ε of Δ_4 exists[ε turns out to be the permutation of Δ_4 associated to ε^'=(0,1,2,3,4).], so that, for each i ∈ℤ_5,  g_ε_i,ε_i+2,ε_i+4= m+1, i.e. t_ε_i,ε_i+2,ε_i+4=0.  So, ρ_ε^'(Γ) - ρ_ε(Γ) =q easily follows from relation (<ref>).

On the other hand,  if  a cyclic permutation ε of Δ_4 exists, so that ρ_ε^'(Γ) - ρ_ε(Γ) =q,  relation (<ref>) yields  t_ε_i,ε_i+2,ε_i+4=0, i.e. g_ε_i,ε_i+2,ε_i+4= m+1, which is exactly - up to color permutation - the definition of weak semi-simple crystallization.
Hence, (a) and (b) are proved to be equivalent.  

Then, by comparing Proposition <ref> (with the assumption g_î=1   ∀ i ∈Δ_4) and relation (<ref>), we obtain:

    ρ_ε^'(Γ) + ρ_ε(Γ) = 2 ( 2 χ(M^4) + 5m -4) + q.

By making use of relation (<ref>),

    ρ_ε(Γ)  = 2 χ(M^4) + 5m -4 + ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4

easily follows, as well as

    ρ_ε^'(Γ)  = 2 χ(M^4) + 5m -4 + ∑_i∈ℤ_5  t_ε_i,ε_i+1,ε_i+2.

Relation (<ref>) directly yields the co-implication between statements (b) and (c).


We conclude the paper with a list of remarks, which arise from the previous results.

 With the above notations, formulae (<ref>) and (<ref>) immediately give

    2 + ρ_ε(Γ)/2 - 5 m/2 - q/4 ≤ χ(M^4)   ≤ 2 + ρ_ε^'(Γ)/2 - 5 m/2 - q/4

for each crystallization  of a closed PL 4-manifold M^4.
Another double inequality concerning the Euler characteristic of a closed  PL 4-manifold M^4 and the regular genera of any order 2p crystallization of M^4 with respect to a pair of associated permutations may be easily obtained from  Proposition <ref>,
by making use of the assumptions ∑_i∈Δ_4  g_î=5 and ρ_ε(Γ) ≤ρ_ε^'(Γ):

    2ρ_ε(Γ) -p +3  ≤ χ(M^4)   ≤ 2 ρ_ε^'(Γ) -p +3.

Note that such double inequalities assume a specific relevance in case of “low" difference between ρ_ε^'(Γ) and ρ_ε(Γ) (in particular if  ρ_ε^'(Γ)= ρ_ε(Γ) occurs, possibly with ρ(Γ) < ρ_ε(Γ)).

On the other hand, Proposition <ref> and formula (<ref>)  (resp. formula (<ref>)) give

    χ(M^4)   =  2 ρ_ε(Γ) -p + 3 + (q -2 ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4)


    (resp.   χ(M^4)   =  2 + ρ_ε(Γ)/2 - 5 m/2 - 1/2∑_i∈ℤ_5   t_ε_i,ε_i+2,ε_i+4 ).

Hence, the following double inequalities arise, too, both involving the regular genus with respect to only one cyclic permutation: 
    2ρ_ε(Γ) -p +3  ≤ χ(M^4)   ≤ 2 ρ_ε(Γ) - p + q +3;


    2 + ρ_ε(Γ)/2 - 5 m/2 - q/4 ≤ χ(M^4)   ≤ 2 + ρ_ε(Γ)/2 - 5 m/2 .





 Note that relation (<ref>) exactly corresponds, via Proposition <ref>, to <cit.>. Moreover, relation (<ref>)[Actually, relation (<ref>) corrects a trivial error in the proof (and statement) of <cit.>, not affecting the implications in order to prove the main result of that paper.]
directly implies the inequality ρ_ε(Γ)  ≥ 2 χ(M^4) + 5m -4 (which is one of the upper bounds obtained in <cit.>) and ensures that, for each crystallization (Γ,γ) of a PL 4-manifold, the regular genus ρ(Γ) is realized by the (not necessarily unique) permutation ε so that ∑_i∈ℤ_5  t_ε_i,ε_i+2,ε_i+4 (or, equivalently, ∑_i∈ℤ_5  g_ε_i,ε_i+2,ε_i+4) is minimal.
Finally, the equivalence between items (b) and (c) in the above Proposition <ref> gives a direct proof of <cit.>.




  Semi-simple crystallizations turn out to be the intersection (characterized by q=0) between the two classes of  weak semi-simple crystallizations and of crystallizations satisfying ω_G(Γ)  =  12 ·ρ (Γ) (and hence ρ_ε(Γ) = 2 χ(M^4) + 5m -4 + 1/2 q, for each cyclic permutation ε of Δ_4).

Moreover, it is easy to check that 
    q≤ 2      ⟹    Γ  is a weak semi-simple crystallization.

In fact, if q = ∑_j,k,l ∈Δ_4 t_j,k,l≤ 2, at most two triads (j,k,l) of distinct elements in Δ_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 Δ_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.



 The formula obtained in <cit.> for bipartite (d+1)-colored graphs and extended to the general case in <cit.> gives, if d=4,

    ω_G(Γ)= 3 (p + 4 - ∑_i∈Δ_4  g_î) + ∑_i∈Δ_4ω_G(Γ_î),

where, for each  i∈Δ_4, ω_G(Γ_î) denotes the sum of the G-degrees of the connected components of Γ_î.
Hence, ∑_i∈Δ_4ω_G(Γ_î) is always a multiple of 3 (recall Proposition <ref> and Theorem <ref>).

Moreover, if (Γ,γ) represents a singular 4-manifold M^4,  Proposition <ref> and Proposition <ref> imply:

    ∑_i∈Δ_4ω_G(Γ_î)   =   3 (2 χ(M^4) + p - ∑_i∈Δ_4  g_î).

Note that, if (Γ,γ) is a crystallization of a closed PL 4-manifold M^4 with rk(π_1(M^4))=m, relation (<ref>) gives:

    ∑_i∈Δ_4ω_G(Γ_î)   =
     3 [5 (χ(M^4) + 2m -2 ) + q ].

In particular, if (Γ,γ) is semi-simple (i.e. q=0), ∑_i∈Δ_4ω_G(Γ_î)  =  15 (χ(M^4) + 2m -2) follows, as <cit.> trivially implies.







§ ACKNOWLEDGMENTS

This work was supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INDAM) and by University of Modena and Reggio Emilia, projects  “Colored graphs representing pseudomanifolds: an interaction with random geometry and physics" and “Applicazioni della Teoria dei Grafi nelle Scienze, nell'Industria e nella Societá".


The authors would like to thank Gloria Rinaldi for her helpful ideas and suggestions about relationship between cyclic permutation properties and Hamiltonian cycle decompositions of complete graphs.

They also thank the referees for their very helpful suggestions.  







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