Generalizations of Baxter's theorem and cubical homology
Consider a 3-regular planar graph. Arbitrarily assign one of four colors to each of its regions. Let \(A_1\) be the number of vertices surrounded by regions of colors 234 in clockwise order, minus the number of vertices surrounded by regions of colors 234 in counterclockwise order. Define \(A_2, A_3, A_4\) similarly. In [J. Comb. Theory 1, 375-384 (1966; Zbl 0144.45201)] \textit{G. Baxter} proved the following result: \[ \text{Theorem 1.} \quad A_1 = - A_2 = A_3 = - A_4. \] In this paper we generalize this theorem to triangulated and cubically subdivided manifolds of arbitrary dimension in several ways. From our cubical results we obtain a way to construct a cubical analogue of simplicial homology.
scientific article; zbMATH DE number 734535
scientific article