A Counting Lemma and Multiple Combinatorial Stokes' Theorem

Europ. J. Combinatorics (1998) 19, 969–979 Article No. ej980247

A Counting Lemma and Multiple Combinatorial Stokes’ Theorem S HYH -N AN L EE AND M AU -H SIANG S HIH† We prove in the present paper a counting lemma for bipartite digraphs which is useful in the combinatorics of pseudomanifolds. By an application of the lemma, we prove a multiple combinatorial Stokes’ theorem, generalizing the 1967 Ky Fan combinatorial formula to multiple labelings. c 1998 Academic Press

1.

I NTRODUCTION

The object of this note will be to prove a counting lemma for bipartite digraphs. By an application of the lemma, we shall establish a ‘Multiple Combinatorial Stokes’ Theorem’. A proof given by Bondy and Murty [1, p. 21] for the Sperner lemma is well-known and makes use of the following simple fact in graph theory: in any (finite) graph, the number of vertices of odd degree is even. On the other hand, in his constructive proof of the Fundamental Theorem of Algebra, Kuhn [3, p. 157] presented the following combinatorial version of Stokes’ theorem (see Figure 1). C OMBINATORIAL S TOKES ’ T HEOREM . Let Q be a bounded region of the plane (connected or not) subdivided into triangles with a positive orientation. Suppose that the vertices of the subdivision are labelled 1, 2, or 3. Define σ (1) = +1 or −1 if the triangle 1 is labelled (1, 2, 3) or (1, 3, 2), in cyclic order, respectively, and σ (1) = 0, otherwise. For a directed edge e ⊂ ∂ Q (∂ Q stands for the boundary of Q), define σ (e) = +1 or −1 if the directed edge e is labelled (1, 2) or (2, 1), respectively, and σ (e) = 0, otherwise. Then X X σ (1) = σ (e). 1⊂Q

e⊂∂ Q

In fact, a more general form of the ‘Combinatorial Stokes’ Theorem’ was proved by Ky Fan [2] (known as Ky Fan’s combinatorial formula), in attempting to give a common generalization of the celebrated Sperner lemma [6] and of the celebrated Tucker lemma [7]. Ky Fan’s formula is only concerned with a single labeling. In the present paper, we shall prove a ‘Multiple Combinatorial Stokes’ Theorem’ and see that this theorem can be proved by using a counting lemma related to bipartite digraphs. Recent research related to combinatorics of complexes may be found in our notes [4, 5]; note [5] is a study of combinatorial formulae for multiple setvalued labelings, with a topological application. It may be mentioned here that our counting lemma for bipartite digraphs also provides a unified treatment of combinatorial theorems proved in [5]. 2.

A C OUNTING L EMMA FOR B IPARTITE D IGRAPHS

We follow the notation given in the book by Bondy and Murty [1]. A digraph (directed graph) D is an ordered triple (V (D), A(D), ψ D ) consisting of a nonempty set V (D) of vertices, a set A(D), disjoint from V (D), of arcs, and an incidence function ψ D that associates with each † This research was supported by the National Science Council of the Republic of China.

0195-6698/98/080969 + 11

$30.00/0

c 1998 Academic Press

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S.-N. Lee and M.-H. Shih

1

–

2

3

–

+

2 + 1

–

+

– 2 2

+

2 3

1

3

+ +

3

3

1

F IGURE 1.

arc of D an ordered pair of (not necessarily distinct) vertices of D. If a is an arc and u and v are vertices such that ψ D (a) = (u, v), then a is said to join u to v; u is the tail of a, and v is its − (v) of a vertex v in D is the number of arcs with head v; the outdegree head. The indegree d D + d D (v) of v is the number of arcs with tail v. A digraph is strict if it has no loops and no two arcs with the same ends have the same orientation. A bipartite graph is one whose vertex set can be partitioned into two sets X and Y , so that each edge has one end in X and one end in Y ; such a partition (X, Y ) is called a bipartition of the graph. A bipartite digraph D means that the underlying graph of D is bipartite. For a set X , denote by |X | the cardinality of X . With the notation explained above, a counting lemma for bipartite digraph runs as follows (see Figure 2). L EMMA . Let p and q be two positive integers and D = (V (D), A(D), ψ D ) be a strict ( f inite) digraph which is bipartite with the bipartition (X, Y ) of the vertex set V (D) of the digraph D. Assume that there exists a partition {X 0 , X 00 } of X and a partition {Y 0 , Y 00 } of Y such that + − + − (v) + d D (v) = p and d D (v)d D (v) = 0, (i) A vertex v of X is in X 0 if d D + − + − 0 (ii) A vertex v of Y is in Y if d D (v) + d D (v) = q and d D (v)d D (v) = 0, + − (v) = d D (v) for all v ∈ X 00 ∪ Y 00 . (iii) d D

Then

p(|X + | − |X − |) + q(|Y + | − |Y − |) = 0,

where X + = {v X − = {v Y + = {v Y − = {v

+ ∈ X 0 : dD (v) = p}, − 0 ∈ X : d D (v) = p}, + ∈ Y 0 : dD (v) = q}, − 0 ∈ Y : dD (v) = q}.

P ROOF. Let X = {x1 , . . . , xr } and Y = {y1 , . . . , ys }, where r = |X | and s = |Y |. Define the incidence matrix (λi j )r ×s of the bipartite digraph D as follows: ( 1 if (xi , y j ) ∈ ψ D (A(D)) and (y j , xi ) ∈ / ψ D (A(D)), λi j = −1 if (y j , xi ) ∈ ψ D (A(D)) and (xi , y j ) ∈ / ψ D (A(D)), 0 otherwise. Then for each i = 1, . . . , r, by conditions (i) and (iii), we have  s p if xi ∈ X + , X + − λi j = d D (xi ) − d D (xi ) = − p if xi ∈ X − ,  j=1 0 if xi ∈ X 00 .

A counting lemma and multiple combinatorial Stokes’ theorem

X0 x1

971

X00

x2

x3

x4

x5

x6

x7

x8

y1

y2

y3

y4

y5

y6

y7

Y0

x9

Y00 F IGURE 2.

Thus

s r X X

λi j = p(|X + | − |X − |).

i=1 j=1

Similarly, we have

r s X X

λi j = −q(|Y + | − |Y − |),

j=1 i=1

2

completing the proof.

For an example, see the digraph shown in Figure 2. This indicates that p = 2, q = 3, and X − = {x1 , x4 }, X + = {x2 , x3 , x5 , x6 , x7 }, + − Y = {y2 , y3 , y4 }. Y = {y1 }, We have

p(|X + | − |X − |) = −q(|Y + | − |Y − |) = 6,

as desired. 3.

A C OMBINATORIAL F ORMULA IN P SEUDOMANIFOLDS

An (abstract) complex is a finite collection K of finite sets such that if σ is an element of K so is every subset of σ . Elements of K are called simplices of K . For each non-negative integer p, let K ( p) be the collection of K such that σ ∈ K ( p) if σ ∈ K and |σ | = p + 1. Elements of K ( p) are called p-simplices of K . A 0-simplex of K is also called a vertex of K . For a p-simplex σ , the closure Cl(σ ) of σ is the complex having all subsets of σ as its elements. Elements of Cl(σ )(r ) are called r -faces of σ (0 ≤ r ≤ p), and elements of Cl(σ ) are called faces of σ . Let A = {a1 , . . . , a p+1 } be a set of p + 1 elements ( p ≥ 1). Then there are ( p + 1)! orderings of the elements of A. Two orderings I = (ai1 , . . . , ai p+1 ) and J = (a j1 , . . . , a j p+1 ) have the same orientation if the permutation   ai1 , . . . , ai p+1 a j1 , . . . , a j p+1

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is an even permutation. We denote this relation by I ∼ J . From elementary properties of permutations, ∼ is an equivalence relation. The set of all orderings (of the elements of A) is thus decomposed into two equivalence classes. Each of the equivalence classes is called an orientation of A, and if we fix one of them arbitrarily, the other one is called the opposite orientation. The orientation of A determined by the ordering (ai1 , . . . , ai p+1 ) will be denoted by [ai1 , . . . , ai p+1 ] and the opposite orientation of [ai1 , . . . , ai p+1 ] will be denoted −[ai1 , . . . , ai p+1 ]. An oriented simplex is a simplex equipped with an orientation. Given a p-simplex σ = {v1 , . . . , v p+1 } of a complex K ( p ≥ 1), we shall write the oriented p-simplex σ by σ = +v1 · · · v p+1

or

σ = −v1 · · · v p+1

if the orientation of σ is [v1 , . . . , v p+1 ] or −[v1 , . . . , v p+1 ] respectively. If σ = + v1 · · · v p+1 is an oriented p-simplex ( p ≥ 1), the induced orientation of the ( p − 1)-face {v1 , . . . , v p } in σ is defined to be [v1 , . . . , v p ] or −[v1 , . . . , v p ] if p is even or odd respectively; when p = 1, the induced orientation of {v1 } in σ = +v1 v2 is, by definition,−[v1 ] (note that the orientation [v1 ] of a one-point set {v1 } is undefined). Thus if σ = εv1 · · · v p+1 is an oriented p-simplex, where ε = ±1 and p ≥ 1, then for each i = 1, 2, . . . , p + 1, σ induces an oriented ( p − 1)-face σi = (−1)i−1 εv1 · · · vi−1 vi+1 · · · v p+1

(∗)

of σ ; when p = 1, the induced oriented 0-faces of σ = ±v1 v2 are σ1 = ±v2 and σ2 = ∓v1 . For the convenience of the reader, we recall that an n-pseudomanifold (n ≥ 0) is a complex K having the following two properties: (M1) Every simplex of K is a face of at least one n-simplex of K . (M2) Each (n − 1)-simplex of K is a face of at most two n-simplices of K . An (n − 1)-simplex σ of an n-pseudomanifold K is called a boundary (n − 1)-simplex of K , if σ is a face of exactly one n-simplex of K . The set of all boundary (n − 1)-simplices of an n-pseudomanifold K is denoted by ∂ K ; when ∂ K = ∅, the empty set, we say that K is a closed n-pseudomanifold. As an example, if σ is an n-simplex of a complex K (n ≥ 1), then Cl(σ ) is an n-pseudomanifold and ∂Cl(σ ) = Cl(σ )(n−1) . An n-pseudomanifold K is coherently oriented if all its n-simplices are so oriented that whenever an (n − 1)-simplex σ is a common face of two distinict n-simplices τ and θ , the orientations of τ and θ induce opposite orientations on σ . Let K be a coherently oriented n-pseudomanifold and 8 be a function which assigns to each vertex v of K an m-tuple (m > n) 8(v) = (ϕ 1 (v), . . . , ϕ m (v)) such that (L1) ϕ i (v) is a nonzero integer for all vertex v of K and for all i = 1, . . . , m, (L2) ϕ i (v1 ) + ϕ j (v2 ) 6 = 0 for the vertices v1 , v2 of every 1-simplex {v1 , v2 } of K and every i, j = 1, . . . , m with i 6 = j. Such a function 8 is called an m-multiple labeling of K . For n + 1 distinct integers j1 , . . . , jn+1 arranged in this order, denote by α+ ( j1 , . . . , jn+1 ) [resp. α− ( j1 , . . . , jn+1 )] the number of those pairs (τ, g) such that

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(A1) τ is an n-simplex of K and g is an injection of τ into {1, . . . , m}, (A2) τ = +v1 · · · vn+1 [resp. τ = −v1 · · · vn+1 ] and ϕ g(vi ) (vi ) = ji

for i = 1, . . . , n + 1.

For every boundary (n − 1)-simplex of K , we consider the induced orientation from that of the unique incident n-simplex of K . For n distinict integers j1 , . . . , jn arranged in this order, β+ ( j1 , . . . , jn ) [resp. β− ( j1 , . . . , jn )] will denote the number of those pairs (σ, f ) such that (B1) σ is a boundary (n − 1)-simplex of K and f is an injection of σ into {1, . . . , m}, (B2) σ = +v1 · · · vn [resp. σ = −v1 · · · vn ] and ϕ f (vi ) (vi ) = ji for i = 1, . . . , n. Finally, we define α = α+ − α− and β = β+ − β− . We are now in a position to prove the ‘Multiple Combinatorial Stokes’ Theorem’. T HEOREM . Let 8 be an m-multiple labeling of a coherently oriented n-pseudomanifold K . Then X {α(−k1 , k2 , . . . , (−1)n+1 kn+1 ) + (−1)n α(k1 , −k2 , . . . , (−1)n kn+1 )} 0
X

= (m − n)

β(k1 , −k2 , . . . , (−1)n−1 kn ).

(∗∗)

0
In particular, if K is a closed n-pseudomanifold, then X α(−k1 , k2 , . . . , (−1)n+1 kn+1 ) 0
= (−1)n−1

X

α(k1 , −k2 , . . . , (−1)n kn+1 ).

0
We remark that when ϕ 1 = · · · = ϕ n = ϕ, then for each boundary (n − 1)-simplex σ of K and for each injection f of σ into {1, . . . , m}, ϕ f (v) (v) = ϕ(v)

for all v ∈ σ,

that is, the integer ϕ f (v) (v) is independent of the choice of such an injection f . Because there are exactly Pnm = m(m −1) · · · (m −n +1) injections of the (n −1)-simplex σ into {1, . . . , m}, β+ ( j1 , . . . , jn ),

β− ( j1 , . . . , jn )

and

β( j1 , . . . , jn )

are multiple of Pnm for any n distinict integers j1 , . . . , jn . Thus the function given by βˆ = β/Pnm m is integer-valued as well. As is integer-valued. Similary, αˆ = α/Pn+1 m = (m − n)Pnm , Pn+1

we have, by (∗∗), X

n+1 {α(−k ˆ kn+1 ) + (−1)n α(k ˆ 1 , −k2 , . . . , (−1)n kn+1 )} 1 , k2 , . . . , (−1)

0
=

X 0
ˆ 1 , −k2 , . . . , (−1)n−1 kn ). β(k

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S.-N. Lee and M.-H. Shih

Thus, when ϕ 1 = · · · = ϕ m = ϕ, the above theorem reduces to Ky Fan’s combinatorial formula [2]. We remark also that in the theorem it is required that injections of an (n + 1)-set into {1, . . . , m}, we must have m > n. If m ≤ n, then both sides of the identity (∗∗) are equal to 0, and there is nothing to discuss. P ROOF OF THE T HEOREM . We divide the proof into four steps, and the proof will be completed by using our counting lemma. Step 1. Construct two sets X and Y . Let X = {(σ, f ) : (σ, f ) satisfies the condition (B1)}, Y = {(τ, g) : (τ, g) satisfies the condition (A1)}. Let (σ, f ) ∈ X . Denote by 8 f (σ ) = {ϕ f (v) (v) : v ∈ σ }. We shall say that (σ, f ) ∈ X is alternating if 8 f (σ ) = {k1 , −k2 , . . . , (−1)n−1 kn } for some integers 0 < k1 < · · · < kn . Step 2. Construct a bipartite digraph D = (V (D), A(D), ψ D ) with the bipartition (X, Y ) of V (D). Define a bipartite strict diagraph D = (V (D), A(D), ψ D ) as follows: (X, Y ) is the bipartition of the vertex set V (D). For x = (σ, f ) ∈ X and y = (τ, g) ∈ Y , the ordered pair (x, y) ∈ ψ D (A(D)) [resp. (y, x) ∈ ψ D (A(D))] if (D1) σ ⊂ τ and f = g|σ, (D2) x is alternating, (D3) τ induces the oriented (n − 1)-face σ = +v1 · · · vn [resp. σ = −v1 · · · vn ] of τ , where ϕ f (vi ) (vi ) = (−1)i−1 ki

for i = 1, . . . , n and 0 < k1 < · · · < kn .

A(D) and ψ D are defined implicitly, and the construction of D is completed. Step 3. Construct four sets X 0 , X 00 , X + , X − . Let x = (σ, f ) ∈ X . + − Case 1. x is not alternating. Then, by (D2), d D (x) = d D (x) = 0. Case 2. x is alternating and σ ∈ / ∂ K . Then, by (M1), (M2) and the definition of ∂ K , σ is a common (n − 1)-face of exactly two distinct n-simplices τ and θ of K . As K is coherently oriented, the orientations of τ and θ induce opposite orientations on σ ; on the other hand, since x = (σ, f ) satisfies condition (B1), there are exactly m − n injective extensions of f to each of the two (n + 1)-sets τ and θ into the set {1, . . . , m}. Thus, by (D1), (D2) and (D3), we + − (x) = d D (x) = m − n. conclude that d D Case 3. x is alternating and σ ∈ ∂ K . Then by the definition of ∂ K , σ is a face of a unique n-simplex τ of K . A similar discussion as in Case 2, we obtain + − (x) = m − n or d D (x) = m − n, dD

and

+ − dD (x)d D (x) = 0.

To fit our counting lemma, we naturally define X 0 = {(σ, f ) ∈ X : (σ, f ) is alternating and σ ∈ ∂ K }, / ∂ K }, X 00 = {(σ, f ) ∈ X : (σ, f ) is alternating and σ ∈ + + 0 X = {x ∈ X : d D (x) = m − n}, − (x) = m − n}. X − = {x ∈ X 0 : d D Note that

|X + | − |X − | =

X 0
β(k1 , −k2 , . . . , (−1)n−1 kn ).

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975

Step 4. Construct four sets Y 0 , Y 00 , Y + , Y − . Let y = (τ, g) ∈ Y . Case 1. There is no alternating x ∈ X such that (x, y) ∈ ψ D (A(D)) or (y, x) ∈ ψ D (A(D)). + − (y) = d D (y) = 0. Then, by (D2), d D Case 2. There is an alternating x = (σ, f ) ∈ X such that (x, y) ∈ ψ D (A(D)), or (y, x) ∈ ψ D (A(D)). Then, by (D1), we may assume that σ = {v1 , . . . , vn } and τ = {v1 , . . . , vn+1 }, and we may write, by (D3) and (∗), σ = εv1 · · · vn , τ = (−1)n εv1 · · · vn+1 (ε = ±1) and ϕ f (vi ) (vi ) = (−1)i−1 ki for i = 1, . . . , n and 0 < k1 < · · · < kn . As f = g|σ by (D1), we have ϕ g(vi ) (vi ) = ϕ f (vi ) (vi ) = (−1)i−1 ki for i = 1, . . . , n. Let ϕ g(vn+1 ) (vn+1 ) = k (k 6 = 0 by (L1)). We discuss the following seven subcases separately. Case 2.1. |k| = ki for some i = 1, . . . , n. Then (by (L2)) .

k = (−1)i−1 ki Thus x 0 = (σ 0 , f 0 ) is alternating, where σ 0 = τ \{vi }

and

f 0 = g|σ 0 .

As the transposition  (vi vn+1 ) =

v1 · · · vi−1 v1 · · · vi−1

vi

vn+1

vi+1 vi+1

· · · vn · · · vn

vn+1 vi



is an odd permutation, we rewrite the oriented n-simplex τ as τ = (−1)n+1 εv1 · · · vi−1 vn+1 vi+1 · · · vn vi and, by (∗), τ induces the oriented (n − 1)-face σ 0 = (−1)εv1 · · · vi−1 vn+1 vi+1 · · · vn of τ . By the definition of A(D) defined implicitly in Step 2 and the hypothesis in this subcase, there is no element in X , other than x and x 0 , joining with y. Consequently, according to (D3), we have + − (y) = d D (y) = 1. dD Case 2.2. ki < |k| < ki+1 for some i = 1, . . . , n − 1 and k = (−1)i−1 |k|. Then by a similar discussion as that in Case 2.1, we can find exactly one alternating x 0 = (σ 0 , f 0 ) other than x such that x 0 and y are joined. Indeed σ 0 = τ \{vi },

f 0 = g|σ 0

and τ induces the oriented (n − 1)-face σ 0 = −εv1 · · · vi−1 vn+1 vi+1 · · · vn . + − (y) = d D (y) = 1 by (D3). Thus d D Case 2.3. ki < |k| < ki+1 for some i = 1, . . . , n − 1 and k = (−1)i |k|. Then, as in Case 2.2 and instead of σ 0 by σ 0 = τ \{vi+1 },

we have

+ − (y) = d D (y) = 1. dD

+ − (y) = d D (y) = 1. Case 2.4. |k| < k1 and k > 0. Then d D

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A

D

B

C F IGURE 3. TABLE 1. ϕ1 ϕ2 ϕ3

A 1 2 3

B −1 2 −3

C −5 −6 7

D 1 6 3

Case 2.5. |k| < k1 and k < 0. Then x is the only one alternating vertex of X such that x and y are joined, therefore + − (y) + d D (y) = 1 dD and



+ (y) = 1 dD − (y) = 1 dD

if ε = −1, if ε = 1.

Case 2.6. |k| > kn and k = (−1)n−1 |k|. Then + − (y) = d D (y) = 1 dD

as in Case 2.4 .

Case 2.7. |k| > kn and k = (−1)n |k|. Then as in Case 2.5, we have + − dD (y) + d D (y) = 1

and



+ (y) = 1 dD − (y) = 1 dD

if ε = −1, if ε = 1.

Let us define Y 0 = {y Y 00 = {y Y + = {y Y − = {y

+ − ∈ Y : dD (y) + d D (y) = 1}, + − ∈ Y : d D (y) + d D (y) 6= 1}, + ∈ Y 0 : dD (y) = 1}, − 0 ∈ Y : d D (y) = 1}.

A counting lemma and multiple combinatorial Stokes’ theorem

X

TABLE 2. β(k1 , −k2 ) = 3 − 4 = −1

0
(σ, f ) (AB, 12) (AB, 13) (AB, 21) (AB, 23) (AB, 31) (AB, 32) (BC, 12) (BC, 13) (BC, 21) (BC, 23) (BC, 31) (BC, 32) (C A, 12) (C A, 13) (C A, 21) (C A, 23) (C A, 31) (C A, 32)

X 0
8 f (σ ) 1 2 1 −3 2 −1 2 −3 3 −1 3 2 −1 −6 −1 7 2 −5 2 7 −3 −5 −3 −6 −5 2 −5 3 −6 1 −6 3 7 1 7 2

Orientation + +

+

− − − −

TABLE 3. X α(−k1 , k2 , −k3 ) = 0 − 1 = −1, (τ, g) (AB D, 123) (AB D, 132) (AB D, 213) (AB D, 231) (AB D, 312) (AB D, 321) (BC D, 123) (BC D, 132) (BC D, 213) (BC D, 231) (BC D, 312) (BC D, 321) (C AD, 123) (C AD, 132) (C AD, 213) (C AD, 231) (C AD, 312) (C AD, 321)

8g (τ ) 1 2 1 −3 2 −1 2 −3 3 −1 3 2 −1 −6 −1 7 2 −5 2 7 −3 −5 −3 −6 −5 2 −5 3 −6 1 −6 3 7 1 7 2

α(k1 , −k2 , k3 ) = 1 − 1 = 0.

0
Orientation 3 6 3 1 6 1 3 6 3 1 6 1 3 6 3 1 6 1

+

−

−

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S.-N. Lee and M.-H. Shih

Note that the permutation  π=

vn+1 v1

v1 v2

· · · vn−1 · · · vn

vn vn+1



has the sign sign(π) = (−1)n , so that by in Case 2.5, we have the following: for the oriented n-simplex τ = εvn+1 v1 · · · vn (= (−1)n εv1 · · · vn+1 ) which induces the oriented (n − 1)-face σ = εv1 · · · vn such that ϕ g(vn+1 ) (vn+1 ) = −|k| , ϕ g(vi ) (vi ) = (−1)i−1 ki

for i = 1, . . . , n, where 0 < |k| < k1 < · · · < kn

Similarly, Case 2.7 implies that for the oriented n-simplex τ = (−1)n εv1 v2 · · · vn+1 , which induces the oriented (n − 1)-face σ = εv1 · · · vn such that ϕ g(vi ) (vi ) = g(v ϕ n+1 ) (vn+1 ) Thus we have X

(−1)i−1 ki for i = 1, . . . , n = (−1)n |k|, where 0 < k1 < · · · < kn < |k|.

{α(−k1 , k2 , . . . , (−1)n+1 kn+1 ) + α(k1 , −k2 , . . . , (−1)n kn+1 )}

0
= |Y − | − |Y + |. Finally, the definitions of X 0 , X 00 in Step 3 and Y 0 , Y 00 in Step 4 imply that (X 0 , X 00 ) and (Y 0 , Y 00 ) are bipartitions of X and Y , respectively. Moreover, conditions (i), (ii) and (iii) are also satisfied with p = m − n and q = 1. Thus (m − n)(|X + | − |X − |) + (|Y + | − |Y − |) = 0, 2

and this identity is equivalent to (∗∗), the proof is complete.

E XAMPLE . Let the 2-pseudomanifold K and 8 = (ϕ 1 , ϕ 2 , ϕ 3 ) be given in Figure 3 and defined in Table 1. Then m = 3, n = 2. From Tables 2 and 3, we have X X {α(−k1 , k2 , −k3 ) + (−1)2 α(k1 , −k2 , k3 )} = (m − n) = β(k1 , −k2 ) = −1. 0
0
A CKNOWLEDGEMENTS The authors are deeply indebted to the referees for their valuable comments. R EFERENCES 1. }J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976. 2. }K. Fan, Simplicial maps from an orientable n-pseudomanifold into S m with the octahedral triangulation, J. Comb. Theory, 2 (1967), 588–602.

A counting lemma and multiple combinatorial Stokes’ theorem

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3. }H. W. Kuhn, A new proof of the fundamental theorem of algebra, Math. Programming Study, 1 (1974), 148–158. 4. }M. H. Shih and S. N. Lee, A combinatorial Lefschetz fixed-point formula, J. Comb. Theory Ser. A, 61 (1992), 123–129. 5. }M. H. Shih and S. N. Lee, Combinatorial formulae for multiple set-valued labellings, Math. Ann., 296 (1993), 35–61. 6. }E. Sperner, Neuer Beweis f¨ur die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Semin. Univ. Hamburg, 6 (1928), 265–272. 7. }A. W. Tucker, Some topological properties of disk and sphere, Proc. First Canadian Math. Congress (Montreal, 1945), University of Toronto Press, Toronto, 1946, pp. 285–309. Received 28 March 1998 and accepted 23 July 1998 S.-N. L EE AND M.-H. S HIH Department of Mathematics Chung Yuan University Chung-Li, Taiwan 32023, Taiwan E-mail: [email protected]