faces of faces of the acyclic birkhoff polytope
TRANSCRIPT
Journal of Mathematical Sciences, Vol. 182, No. 2, April, 2012
FACES OF FACES OF THE ACYCLIC BIRKHOFF POLYTOPE
L. Costa and E. A. Martins UDC 519.17
Abstract. Given a p-face of the acyclic Birkhoff polytope Ωn(T ), where T is a tree with n vertices,we find the number of faces of lower dimension that are contained in it, and its nature is discussed.
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442. Number of Different Configurations of the T -components in a Face of Ωn(T ) . . . . . . . 1453. Comparing Faces of Ωn(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454. Faces of Lower Dimension of a Given 2-Face of Ωn(T ) . . . . . . . . . . . . . . . . . . . . 1485. Faces of Lower Dimension of a Given Cell of Ωn(T ) . . . . . . . . . . . . . . . . . . . . . . 1516. Number of Faces of a Given p-Face of Ωn(T ) . . . . . . . . . . . . . . . . . . . . . . . . . 155
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
1. Introduction
The main motivation for this work stems from a recent publication [7] that deals with the countingof the number of faces of faces of the tridiagonal Birkhoff polytope, Ωt
n, that is, the set of real squarematrices with nonnegative entries and all sums of elements of each row or column equal to one, thatare tridiagonal. In fact, for a given p-face of Ωt
n, we determine the number of faces of lower dimensionthat are contained in it, and we study its nature. The above subject can also be assigned to the acyclicBirkhoff polytope, Ωn(T ), that is, the set of matrices whose support corresponds to some subset ofthe edges (including loops) of a fixed tree T with n vertices.
In [5] we saw that for a face of any dimension, there is a one-to-one correspondence between it andthe union of a finite number of bicolored subgraphs of one of the three following types:
(1) a closed vertex • ;(2) an open edge ◦ ◦ ;(3) this type is not one of the previous two types and is a bicolored subgraph obtained from any
connected bicolored subgraph of T with all endpoints closed.
It was also given the name of T -component and inner entry.In [7], it was established the maximum number of T -components that can be present in the bicolored
subgraph that represents a p-face, with p ≥ 2, of Ωtn. We also study the number of vertices associated
to a p-face of the referred polytope and we present the number of faces of lower dimension of a given2-face and of a given cell of Ωt
n. Note that a face of Ωtn of dimension greater than 0 has always a
T -component that can be seen as a path. The counting of its vertices was done considering differentconfigurations, and the approach was made in a combinatorial way using Fibonacci numbers. Somealgorithms for its counting were given. Recall that that the expression
gm(T ∪ T ′) =m∑
k=0
fm−k(T )fk(T′) (1.1)
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica-tions), Vol. 71, Algebraic Techniques in Graph Theory and Optimization, 2011.
144 1072–3374/12/1822–0144 c© 2012 Springer Science+Business Media, Inc.
gives, when m = 0, the number of vertices of the union of the connected components T and T ′, evenwhen these last ones are T -components.
The scheme of this paper is as follows.Section 2 contains a result that gives the maximum number of different possibilities in the con-
struction of the bicolored subgraph that represents a p-face, considering all the different numbers ofT -components.
In Sec. 3, we discuss some comparing results related with the faces of Ωn, namely, for a given pairof faces with different dimensions we present an inclusion relation between them, and in the case ofequal dimensions we discuss its possible adjacency.
In Sec. 4, the number of faces of lower dimension of a given 2-face is determined, while in Sec. 5,the number of faces of lower dimension that is contained in a given cell is discussed. Finally, in thelast section we present the number of vertices and the number of (p− 1)-faces that are contained in agiven p-face of Ωn(T ).
2. Number of Different Configurations of the T -components in a Face of Ωn(T )
From [5], the configuration of a p-face of Ωn(T ) is the union of closed vertices, open edges and tT -components with ι inner entries and θ closed endpoints. Regarding the calculation of the dimensionof a p-face it is only important to consider the T -components and it is given by the following formula:
p = θ + ι− t.
In this section, we examine all possibilities in the construction of a bicolored subgraph that repre-sents a p-face for all t. From the previous formula, as θ ≥ 2t, we obtain t ≤ p. Although in the bicoloredsubgraph, there may be internal open vertices, we do not consider it. For each t ∈ {1, . . . , p}, the pointswhose first coordinate is θ and the second coordinate is ι: (2t, p−t), (2t+1, p−t−1), . . . , (2t+(p−t), 0)lie on the straight line θ + ι = p+ t and the number of these points is equal to the number of all thepossibilities for the construction of the p-face. Therefore, for each t, we have p− t+1 possibilities for(θ, ι), called admissible pairs. The next proposition gives the total number of different configurationsof p-faces.
Proposition 2.1. The total number of different configurations to obtain a p-face is
p+ (p− 1) + (p− 2) + · · ·+ 1 =(1 + p)p
2.
Example 2.1. If p = 4, we have, from Proposition 2.1, that the total number of different configura-tions of T -components to obtain a 4-face is ten and the cases are described below:
(i) t = 1; θ+ι = 5; (2, 3), (3, 2), (4, 1), and (5, 0) are the admissible pairs. We have four possibilities.(ii) t = 2; θ + ι = 6; (4, 2), (5, 1), and (6, 0) are the admissible pairs. We have three possibilities.(iii) t = 3; θ + ι = 7; (6, 1) and (7, 0) are the admissible pairs. We have two possibilities.(iv) t = 4; θ + ι = 8; (8, 0) is the only admissible pair. We have one possibility.
3. Comparing Faces of Ωn(T )
The aim of this section is to introduce some comparing results related with the faces of Ωn(T ). Inparticular, given two faces, Fi and Fj , i �= j, we intend to know if there is an inclusion relation betweenthem if dimFi �= dimFj ; otherwise, if there is an adjacency relation. We also define intersection ofany two faces.
Let G1 = (V (G1), E(G1)) and G2 = (V (G2), E(G2)), where E(G1) and E(G2) are subsets of theset of edges of a graph G, and in V (G1) and V (G2) some of the vertices can be closed, that is,
V (G1) = V 1• ⊕ V 1
◦ , V (G2) = V 2• ⊕ V 2
◦ ;
G1 and G2 are bicolored subgraphs, and its sum is the bicolored subgraph of G such that
G1 �G2 =(V◦ ⊕ V•, E(G1) ∪ E(G2)
),
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whereV◦ = V 1
◦ ∩ V 2◦ , V• = V 1
• ∪ V 2• ,
i.e.,◦1 � ◦2 = ◦,•1 � ◦2 = •,◦1 � •2 = •,•1 � •2 = •,
where ◦1 and ◦2 and •1 and •2 denote, respectively, the open and closed vertices of the bicoloredsubgraphs G1 and G2 (see [5]). We also set up a criterion for the adjacency of the vertices of theacyclic Birkhoff polytope Ωn(T ).
Now we define strictly inclusion, intersection, and adjacency of a pair of faces of Ωn(T ).
Definition 3.1. Given two faces Fi and Fj , i �= j, we say that Fi is strictly included in Fj , Fi ⊂ Fj ,if
dimFi < dimFj
and there is a face F such that
dimF < dimFj , Fi � F = Fj .
Example 3.1. Let F1 and F2 be the following faces:
F1 : ◦ ◦ • • F2 : ◦ ◦ ◦ ◦as dimF2 = 0 < dimF1 = 1 and the face
F : ◦ ◦ • •is such that F2 � F = F1, we say that F2 is strictly included in F1.
Definition 3.2. Given two faces Fi and Fj , i �= j, the intersection of Fi and Fj , Fi ∩Fj , is the faceF such that
F ⊂ Fi, F ⊂ Fj ,
we have
dimF ≤ min{dimFi, dimFj
}
and for all faces F ′ such thatF ′ ⊂ Fi, F ′ ⊂ Fj ,
we havedimF ′ < dimF ;
otherwiseFi ∩ Fj = ∅.
Example 3.2. Let F1 and F2 be the following faces of Ω4:
F1 : ◦ ◦ • • F2 : • • ◦ ◦as dimF1 = dimF2 = 1 and the face
F : ◦ ◦ ◦ ◦has dimension 0 and is strictly included in F1 and in F2, F1 ∩ F2 = F .
Consider F1 as above and
F3 : • • • • , dimF3 = 1,
and there are no faces included simultaneously on both of them; therefore,
F1 ∩ F3 = ∅.
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Definition 3.3. Given two faces of the same dimension greater than 0, Fi,Fj , i �= j, we say that Fi
is adjacent to Fj if there is a face F such that
dimF = dimFj − 1, Fi ∩ Fj = F ;
otherwise, we say that the faces are nonadjacent.
In the next example we present faces that are adjacent and faces that are nonadjacent.
Example 3.3. Let S′ = S1,2,3 be the starlike tree with three branches of lengths 1, 2, and 3, respec-tively, and F1, F2, F3, F4, and F5 be the five following faces of Ω(S′) presented below:
F1 : • ◦ • • •••
F2 : ◦ ◦ • • •••
F3 : • ◦ ◦ • •••
F4 : ◦ ◦ • • •◦◦
F5 : • ◦ • • •••
Here dimF1 = 2 and dimF2 = dimF3 = 1. We also have
F2 � F3 = F1.
Therefore,
F2 ⊂ F1, F3 ⊂ F1.
Observe that we also have
dimF3 = dimF4 = 1, F3 ∩ F4 = ∅,and we conclude that the faces F3 and F4 are nonadjacent. Since
dimF2 = dimF4 = 1,
and there exists a face F , represented below, such that
F2 ∩ F4 = F, dimF = dimF2 − 1
it follows that the faces F2 and F4 are adjacent:
F : ◦ ◦ • • •◦◦
For a given p-face F , to determine the faces of Ωn(T ) that are adjacent to it, we must consider allthe faces F ′ of dimension p and for each pair F , F ′ we do the respective intersection and verify if thedimension of the resultant face is p− 1.
To determine the intersection of two given faces, we proceed with the following steps:
(1) determine the vertices of each face;(2) choose the common vertices between the faces;(3) if there are no common vertices, then the intersection is the empty set; otherwise do the bicolored
sum of the respective common vertices and the ouput is the intersection.
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4. Faces of Lower Dimension of a Given 2-Face of Ωn(T )
In this section, we find the number of vertices and edges that are contained in a given 2-face ofΩn(T ) and describe their representations in terms of bicolored subgraphs.
Throughout the text, Hi, where i is a positive integer, denotes the union of a finite number ofbicolored subgraphs of types 1 and 2; H0 denotes the empty set, and in this case, we assume thatf0(∅) = 1; in a bicolored subgraph, the number of internal vertices is ω = η + ι ≥ 0, where η is thenumber of internal open vertices.
Regarding that the number of T -components in a configuration of a 2-face is at most 2, we displayall the possible cases.
(1) one T -component with two closed endpoints and one inner entry. In this case, the inner entrycan occupy all the positions corresponding to the internal vertices, for example,
H1 • • ◦ ◦ ◦ • H2.
(2) One T -component with three closed endpoints and without inner entries. The configuration isas follows:
H1 •���
•���◦ ◦
◦◦•
H2
H3.
(3) Two T -components each one with two closed endpoints and without inner entries. The bicoloredsubgraph is represented below:
H1 • ◦ ◦ • H2 • ◦ ◦ • H3.
In the next subsection, we characterize the vertices that are contained in the 2-face when it is describedas in cases 1–3.
4.1. Number of vertices of a 2-face. Bicolored subgraphs with two closed endpoints describedin cases 1 and 3 can be considered as “paths,” and the number of vertices associated to these config-urations has already been discussed in [7]. In fact, for case 1, due to [7, Propositions 3.1 and 3.3], thereferred number associated to the face that has only a T -component with such configuration is always3. For case 3, from [7, Propositions 3.1 and 3.5], the number of vertices of a p-face with the describedconfiguration is f0(P2)f0(P2) = 4.
Now we will focus our attention on the remaining case.First, we consider the simplest case where η = 1. We have the following three vertices obtained as
in [5]:
V1 : H1 ◦ ◦ ••
H2
H3
V2 : H1 • ◦ ◦•
H2
H3
V3 : H1 • ◦ •◦
H2
H3
Note that these vertices are also obtained if we consider the three edges of the same face and do itsintersection two by two:
e1 : H1 • ◦ ••
H2
H3
e2 : H1 • ◦ ••
H2
H3
e3 : H1 • ◦ ••
H2
H3
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The first two edges, e1 and e2, have the common vertex V3. The first and the third edges, e1 and e3,have the common vertex V1 and finally, the two last ones, e2 and e3, have the common vertex V2.
If η > 1, we proceed in a similar way and we have three possible situations that are similar tothe previous ones. Each edge ei, i = 1, 2, 3, is represented by the union of a T -component with twoendpoints and without inner entries, and bicolored subgraphs of type 1 and type 2:
e1 : H1 •���◦
◦◦•
H2
H3
e2 : H1 •���
•���◦ ◦
◦
H2
H3
e3 : H1 •���
◦◦◦•
H2
H3
Again, each pair of edges has a common vertex. Therefore, the number of vertices of this face is 3.
Proposition 4.1. The number of vertices (edges) of a 2-face of Ωn(T ) is equal to the sum of allclosed endpoints and the number of inner entries present in all T -components of its configuration.
Bearing in mind that in cases 1 and 2, the number of vertices is 3, and in case 3, the number ofvertices is 4, we can establish the following assertions.
Corollary 4.2. The faces of Ωn(T ) are quadrilateral or trilateral.
Recall that bicolored subgraphs of 2-faces of Ωn(S), where S is a star, can have only T -componentsas described in cases 1 and 2. Then we obtain the following.
Corollary 4.3. The faces of Ωn(S) are trilateral.
Proposition 4.4. Every acyclic Birkhoff polytope whose dimension is greater than 2 has at least onetriangular face.
Next, we present some results derived from the previous ones. First, we illustrate with an example.An analogous description has already been made in [7, Sec. 6]; however, in the present case, we use astarlike tree, not a path.
Example 4.1. Let S′ = S1,2,3 be the starlike tree
◦ ◦ ◦ ◦ ◦◦◦
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One 2-face of Ω7(S′) is
F : • • • • •••
The vertices that are contained in this face are
• • • • •••
◦ ◦ • • •••
V1 V2
• • ◦ ◦ •••
◦ ◦ ◦ ◦ •••
V3 V4
SinceV1 � V4 = V2 � V3 = F ,
where F is a 2-face, we say that V1 and V4 are opposite vertices and that V2 and V3 are also opposite.We can conclude than the face is quadrilateral.
The next two propositions give a generalization for the acyclic Birkhoff polytope of [7, Propositions 6.1and 6.2] for Ωt
n, the tridiagonal Birkhoff polytope.
Proposition 4.5. Let Vi and Vj, i �= j, be two vertices of a 2-face of Ωn(T ), F . If Vi � Vj = F , thenVi and Vj are nonadjacent (they are opposite vertices) and the face is a quadrilateral.
Proposition 4.6. Let Vi and Vj, i �= j, be two vertices of a 2-face of Ωn(T ). Then Vi and Vj areadjacent or opposite.
Example 4.2. Consider the following vertices from Ω7(S′):
• • • • •◦◦
and ◦ ◦ • • •••
If we consider their bicolored sum, we obtain the bicolored subgraph
• • • • •••
This bicolored subgraph is the configuration of a 2-face; this means that those vertices are oppositeand the face is a quadrilateral.
From this last configuration we can obtain the two remaining vertices of the face, namely,
◦ ◦ • • •◦◦
and • • • • •••
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Note that these last ones are also opposite vertices.
Regarding the nature and number of the configuration of the T -components that belongs to theconfiguration of a face, we can state the following assertion.
Proposition 4.7. Let T be a tree that has at least a path P4 as a subgraph. Then Ωn(T ) has at leasta quadrilateral face.
4.2. Number of edges of a 2-face. The number of vertices of a 2-face is the same as its numberof edges. As a consequence of the study made in the previous subsection, we conclude that a 2-facecan have 3 or 4 edges.
5. Faces of Lower Dimension of a Given Cell of Ωn(T )
In this section, we present the number of vertices, edges, and faces contained in a cell of Ωn(T ).The number of T -components that can be present in the bicolored subgraph that represents a 3-faceis at most 3. The different cases are described below (see Proposition 2.1):
(i) t = 1; θ + ι = 4; the admissible pairs are (2, 2), (3, 1), and (4, 0);(ii) t = 2; θ + ι = 5; (4, 1) and (5, 0) are the admissible pairs;(iii) t = 3; θ + ι = 6; and (6, 0) is the only admissible pair.
Consider the following situations:
(1) t = 1 and the admissible pair is (2, 2);(2) t = 2 and the admissible pair is (4, 1);(3) t = 3 and the admissible pair is (6, 0).
In these situations, we can consider the T -components as paths and this was studied in [7]. In thesame paper, it was also shown that if we are in case (1), we can have a cell with four vertices orwith five vertices, depending on the number and position of the internal open vertices presented inthe T -component. If the cell has four vertices, it has six edges and four 2-faces, each of them is atrilateral, and the cell is a tetrahedron; if the cell has five vertices, it has eight edges and five 2-faces:four trilateral and one quadrilateral, and the cell is a pentahedron. For this issue, the reader is referredto [7]. In case (2), independently of the number of the internal open vertices that there exist in eachT -component, we have 3 × 2 = 6 vertices of the cell. These vertices give origin to nine edges, whichcorrespond to five faces: two of these faces are trilateral and three of them are quadrilateral. This cellis a pentahedron. In case (3), again, independently of the number of the internal open vertices thatcan be present in each T -component, each of them corresponds to two configurations. Therefore, wehave 2 × 2 × 2 = 8 vertices. These vertices lead to twelve edges that correspond to six faces, all ofthem are quadrilateral, and the cell is a hexahedron.
The remaining cases are as follows:
(4) t = 1 and (3, 1), (4, 0) are the admissible pairs;(5) t = 2 and the admissible pair is (5, 0).
In case (4), the bicolored subgraph has one T -component with three closed endpoints and one innerentry.
If η = 0,
H1 • • ••
H2
H3
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The cell contains the next four vertices:
H1 • • ••
H2 H1 • ◦ ◦•
H2
H3 H3
H1 ◦ ◦ ••
H2 H1 • ◦ •◦
H2
H3 H3
It has the following six edges:
H1 • • ••
H2 H1 • • ••
H2
H3 H3
H1 • ◦ ••
H2 H1 • • ••
H2
H3 H3
H1 • ◦ ••
H2 H1 • ◦ ••
H2
H3 H3
and it has the four 2-faces:
H1 • • ••
H2 H1 • • ••
H2
H3 H3
H1 • • ••
H2 H1 • ◦ ••
H2
H3 H3
The cell is a tetrahedron.If η > 0
and the vertex with maximum degree is closed, the situation described above still remains, andthe cell has four vertices, six edges and four 2-faces, and it is a tetrahedron;
η = 1 and the vertex with maximum degree is open:
H1 • ◦ •••
H2
H3
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The cell contains the following five vertices:
H1 • ◦ ◦••
H2 H1 • ◦ •◦•
H2 H1 ◦ ◦ •••
H2
H3 H3 H3
H1 ◦ ◦ •◦◦
H2 H1 • ◦ ◦◦◦
H2
H3 H3
It has the following eight edges:
H1 • ◦ •••
H2 H1 • ◦ •••
H2 H1 • ◦ ◦••
H2
H3 H3 H3
H1 • ◦ •••
H2 H1 • ◦ •◦•
H2 H1 • ◦ •◦•
H2
H3 H3 H3
H1 ◦ ◦ •••
H2 H1 • ◦ •◦◦
H2
H3 H3
and it has the following five 2-faces:
H1 • ◦ •••
H2 H1 • ◦ •••
H2 H1 • ◦ •••
H2
H3 H3 H3
H1 • ◦ •••
H2 H1 • ◦ •◦•
H2
H3 H3
Therefore, this cell is a pentahedron (four faces are trilateral and one face is a quadrilateral).
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If η > 1 and the vertex with maximum degree is open, the situation described above still remainsand the cell has five vertices, eight edges, and five 2-faces; therefore, it is a pentahedron.
Now we must consider the other situation present in case (4): if the bicolored subgraph has oneT -component with four closed endpoints and without inner entries.
First, assume that η = 1, the configuration of the cell is
H1
H2 •• ◦ •
•
H3
H4
the cell contains the following four vertices:
H1 H1
H2 •◦ ◦ •
•
H3 H2 ◦• ◦ •
•
H3
H4 H4
H1 H1
H2 •• ◦ ◦
•
H3 H2 •• ◦ •
◦
H3
H4 H4
and the next six edges:
H1 H1 H1
H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3
H4 H4 H4
H1 H1 H1
H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3
H4 H4 H4
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and the four 2-faces:
H1 H1
H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3
H4 H4
H1 H1
H2 •• ◦ •
•
H3 H2 •• ◦ •
•
H3
H4 H4
Therefore the cell is a tetrahedron.If η > 1, the situation described above still remains and the cell has four vertices, six edges, and
four 2-faces; it is a tetrahedron.Finally, consider case (5). The bicolored subgraph has two T -components as follows:
H1 • ◦ ••
H2 • • H3
H4
Regardless of the number of open vertices, three configurations of vertices corresponds to the firstT -component and two configurations of vertices corresponds to the second T -component. Therefore,we have 3 × 2 = 6 vertices. These six vertices give rise to five faces: two of them are trilateral andthree of them are quadrilateral.
Proposition 5.1. The 3-faces of the polytope Ωn(T ) can only be:
(i) tetrahedrons whose 2-faces are trilateral ;(ii) pentahedrons with four trilateral faces and one quadrilateral face or two trilateral faces and three
quadrilateral faces ;(iii) hexahedrons with six quadrilateral faces.
Note that the number of 2-faces of a 3-face depends on the number of closed endpoints and innerentries but in certain cases also depends on the position of the internal open vertices in the bicoloredsubgraph that represents the 3-face.
6. Number of Faces of a Given p-Face of Ωn(T )
In this section, we find the number of vertices and the number of faces of dimension p− 1 that arecontained in a given p-face, F .
6.1. Number of vertices of a given p-face of Ωn(T ). First, we consider the case where thebicolored subgraph that represents F has one T -component. For t = 1, the number of vertices of theT -component is p + 1 + η. Furthermore, the tree with the same vertices of the T -component will bedenoted by Tp+1+η.
If η = 0, f0(F) = f0(Tp+1).If η = 1, the T -component has p + 2 vertices. Let i1 be the internal open vertex whose degree is
k1; T\i1 represents the bicolored subgraph, which is obtained from the T -component by removing thevertex i1, as well as the edges incident on it. We denote by i1j , j = 1, . . . , k1, the adjacent vertices
155
to i1; each of the components of T\i1 is connected and we assign the component that has i1j as aclosed vertex to Ti1j .
The vertices of Ω(Tp+2) that have in their representation the vertex i1 as closed are not vertices ofF . This number is g0(T\i1). The remaining vertices of Ω(Tp+2) are vertices of F . To count these lastones, we proceed as follows:
(i) f0(F) = f0(Tp+2)− g0(T\i1), where
g0(T\i1) =k1∏
j=1
f0(Ti1j );
(ii) therefore,
f0(F) = f0(Tp+2)−k1∏
j=1
f0(Ti1j ).
If η = 2, let ip, p = 1, 2, be the internal open vertex whose degrees are kp; T\ip, represents thebicolored subgraph that is obtained from the T -component by removing the vertex ip, as well asthe edges incident on it. We denote by ipj , j = 1, . . . , kp, the adjacent vertices to ip; each of thecomponents of T\ip is connected, and we assign the component that has ipj as a closed vertex to Tipj .
The vertices of Ω(Tp+3) that have in their representation the vertex i1 or the vertex i2 as closedare not vertices of F ; the remaining vertices of Ω(Tp+3) are vertices of F . To count these last ones weproceed as follows:
f0(F) = f0(Tp+3)− g0(T\i1)− g0(T\i2) + g0(T\{i1, i2}),where
g0(T\i1) =k1∏
j=1
f0(Ti1j ), g0(T\i2) =k2∏
l=1
f0(Ti2l),
and the last summand is also obtained using (1.1).If η = 3, using a similar procedure and appropriate notation we obtain the following expression for
the number of vertices of F :
f0(F) = f0(Tp+4)− g0(T\i1)− g0(T\i2)− g0(T\i3)+ g0(T\{i1, i2}) + g0(T\{i1, i3}) + g0(T\{i2, i3})− g0(T\{i1, i2, i3}).
Proceeding in the same way, we can determine f0(F) for any η.As the number of T -components can be greater than 1, we use the previous procedure for each of
them and finally use the expression (1.1) to obtain the final number.
6.2. Number and construction of (p−1)-faces of a given p-face of Ωn(T ). In this subsection,we determine the number of faces of dimension p− 1 that belongs to a given p-face. It follows that inthe configuration of a p-face we can have at most p(p+ 1)/2 T -components.
If t = 1, the number θ of closed endpoints and the number ι of inner entries are such that ι+θ = p+1,(θ ≥ 2). As we want to construct bicolored subgraphs that corresponds to faces of dimension p − 1,either the number of inner entries or the number of endpoints must decrease by one unit; consequently,we can proceed as follows:
Replace each inner entry by an open vertex, therefore we have ι different possibilities.
or
For each endpoint:� if its neighbor is a closed vertex, remove the endpoint and the edge incident on it andreplace both by a closed vertex;
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� if its neighbor is open and it belongs to a sequence of length even of k internal open vertices,each of them of degree 2, replace the endpoint and the sequence of internal open verticesby a closed vertex and k
2 open edges;� if neighbor is open and it belongs to a sequence of length odd of k internal open vertices,each of them of degree 2, replace the closed endpoint and the k internal open vertices by(k + 1)/2 open edges (recall that in the end we must obtain a T -component);
� if its neighbor is open and it belongs to a sequence of open vertices in which at most one ofthem has degree greater than 2, consider the vertex i of degree greater than 2 closest to theendpoint and replace the sequence of the closed endpoint and the open vertices pendant toi by— a closed vertex and open edges if the number of open vertices until i is even;— open edges if the number of open vertices until i is odd.
Finally, for each endpoint we obtain a face of dimension p− 1; therefore, we have θ differentpossibilities.
In conclusion, we have ι+ θ, i.e., p+ 1 faces of dimension p− 1.If t = 2, let θp, ιp be the numbers of closed endpoints and inner entries, respectively, of the T -
component Tp (p = 1, 2). Now we have
(ι1 + θ1) + (ι2 + θ2) = p+ 2.
Proceeding, in each T -component, similarly to the former way, we can replace each inner entry byan open vertex and therefore we have ι1 + ι2 different possibilities.
We can also replace each closed endpoint as we described.
• If ι1, ι2 ≥ 1, we have θ1 + θ2 possibilities. In this situation, we have ι1 + ι2 + θ1 + θ2, i.e, p+ 2faces of dimension p− 1.
• If ι1 = 0 or ι2 = 0, without loss of generality, let ι1 = 0, θ1 ≥ 2, and ι2 > 0. In this case, weproceed as described above and we have θ1 + ι2 + θ2 faces of dimension p − 1. Therefore, wealso have ι1 + ι2 + θ1 + θ2 = p+ 2 faces of dimension p− 1.
If t = �, let θp and ιp be the numbers of closed endpoints and inner entries, respectively, of theT -component Tp (p = 1, . . . , �). We have
(ι1 + θ1) + · · ·+ (ι� + θ�) = p+ �.
In each T -component, we can replace each inner entry by an open vertex. From this we obtainι1 + · · · + ι� different possibilities. We can also replace each closed endpoint as described in the firstsituation, and it results in θ1+· · ·+θ� possibilities. In the present case, we obtain ι1+· · ·+ι�+θ1+· · ·+θ�,i.e., p+ � faces of dimension p− 1.
Proposition 6.1. The number of (p − 1)-faces that belong to a p-face of Ωn(T ) is p + k, wherek = 1, . . . , p is the number of T -components present in the configuration of the p-face.
Acknowledgments. This research was partially supported by the Centre for Research and Develop-ment in Mathematics and Applications (CIDMA) from the Fundacao para a Ciencia e a Tecnologia(FCT), cofinanced by the European Community Fund FEDER/POCI 2010.
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Liliana CostaDepartment of Mathematics, University of Aveiro, Aveiro, PortugalE-mail: [email protected] Andrade MartinsDepartment of Mathematics, University of Aveiro, Aveiro, PortugalE-mail: [email protected]
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