Summary: Proof: Let S be the number of complete (m \Gamma 1)­simplexes in K m (colored with f0; : : : ; m \Gamma
1g), where the (m \Gamma 1)­simplexes in each m­simplex are considered separately, and counted as
+1 or \Gamma1, by their induced orientations. We argue that S = I and S = C.
To prove that S = I , consider the following cases. If an (m \Gamma 1)­face is internal, then it
contributes 0 to S, since the contributions of the two m­simplexes containing it cancel each
other. Obviously, an internal (m \Gamma 1)­face contributes 0 to I . An external (m \Gamma 1)­face in the
boundary of K m is counted the same, +1 or \Gamma1 by orientation, in both S and I . Therefore,
S = I .
To prove that C = S, consider an m­simplex ø m , and look at the following cases. If ø m
contains two (m \Gamma 1)­faces which are completely colored, then ø m is not completely colored and
contributes 0 to C. Note that ø m contributes 0 also to S, since the contributions of the two
faces cancel each other. If ø m contains exactly one (m \Gamma 1)­face which is completely colored
(with f0; : : : ; m \Gamma 1g), then ø m must be completely colored and contributes +1 or \Gamma1, by
orientation, to C as well as to S. If ø m does not contain any (m \Gamma 1)­face which is completely
colored, then ø m is not completely colored and therefore, it contributes 0 to C as well as to
S. Finally, note that ø m cannot contain more than two (m \Gamma 1)­faces which are completely
colored.
We can now derive the oriented version of Sperner's Lemma.
Theorem C.1 (Oriented Sperner's Lemma) Consider an orientable divided image K m of
M(oe m ) under /, and a Sperner coloring Ø : K m ! M(oe m ). There exists an odd number of

Source: Attiya, Hagit - Department of Computer Science, Technion, Israel Institute of Technology

Collections: Computer Technologies and Information Sciences