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Keywords:
colouring multidimensional maps; four colour theorem; chromatic number; tetrahedralization; convex polytopes; finite element methods; domain decomposition methods; parallel programming; combinatorial geometry; six colour conjecture
colouring multidimensional maps; four colour theorem; chromatic number; tetrahedralization; convex polytopes; finite element methods; domain decomposition methods; parallel programming; combinatorial geometry; six colour conjecture
Summary:
We consider face-to-face partitions of bounded polytopes into convex polytopes in $\mathbb{R}^d$ for arbitrary $d\ge 1$ and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed $d+1$. Partitions of polyhedra in $\mathbb{R}^3$ into pentahedra and hexahedra are $5$- and $6$-colourable, respectively. We show that the above numbers are attainable, i.e., in general, they cannot be reduced.
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