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MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 251-264 (2002)
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# Colouring polytopic partitions in $\Bbb R^d$

## Michal Krizek

* Michal Krizek*, Mathematical Institute, Academy of Sciences, Zitna 25, CZ-115 67 Praha 1, Czech Republic, e-mail: ` krizek@math.cas.cz`

**Abstract:**
We consider face-to-face partitions of bounded polytopes into convex polytopes in $\Bbb R^d$ 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 $\Bbb R^3$ 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.

**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

**Classification (MSC2000):** 05C15, 51M20, 65N30

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