Colouring polytopic partitions in d

Michal Křížek

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 251-264
  • ISSN: 0862-7959

Abstract

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We consider face-to-face partitions of bounded polytopes into convex polytopes in d for arbitrary d 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 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.

How to cite

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Křížek, Michal. "Colouring polytopic partitions in $\mathbb {R}^d$." Mathematica Bohemica 127.2 (2002): 251-264. <http://eudml.org/doc/249039>.

@article{Křížek2002,
abstract = {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.},
author = {Křížek, Michal},
journal = {Mathematica Bohemica},
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},
language = {eng},
number = {2},
pages = {251-264},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Colouring polytopic partitions in $\mathbb \{R\}^d$},
url = {http://eudml.org/doc/249039},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Křížek, Michal
TI - Colouring polytopic partitions in $\mathbb {R}^d$
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 251
EP - 264
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/249039
ER -

References

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  1. Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), 711–712. (1976) MR0424602
  2. Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977), 429–490. (1977) MR0543792
  3. Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977), 491–567. (1977) MR0543793
  4. PLTMG: A Software Package for Solving Partial Differential Equations: Users’ Guide 7.0, SIAM, Philadelphia, 1994. (1994) MR1262123
  5. Coloring polyhedral manifolds, Proc. Conf. Discrete Geometry and Convexity, Ann. N. Y. Acad. Sci. 440 (1985), 192–195. (1985) Zbl0571.05018MR0809206
  6. Superconvergence and a posteriori error estimation for triangular mixed finite elements, Numer. Math. 68 (1994), 311–324. (1994) Zbl0823.65103MR1313147
  7. The Four-Color Theorem, Springer, Berlin, 1998. (1998) MR1633950
  8. Topological Graph Theory, John Wiley & Sons, New York, 1987. (1987) MR0898434
  9. Grötzch’s theorem on 3-coloring, Michigan Math. J. 10 (1963), 303–310. (1963) MR0154274
  10. Graph Theory, Addison-Wesley, London, 1972. (1972) MR0256911
  11. Map colour theorems, Quart. J. Math. 24 (1890), 332–338. (1890) 
  12. An equilibrium finite element method in three-dimensional elasticity, Apl. Mat. 27 (1982), 46–75. (1982) MR0640139
  13. Finite Element Approximation of Variational Problems and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics vol. 50, Longman Scientific & Technical, Harlow, 1990. (1990) MR1066462
  14. Finite Element Methods: Superconvergence, Postprocessing, and A Posteriori Estimates, Proc. Conf., Jyväskylä, 1996, LN in Pure and Appl. Math. vol. 196, Marcel Dekker, New York, 1998. (1998) MR1602809
  15. Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 217–283. (1930) 
  16. Three short proofs in graph theory, J. Combin. Theory Ser. B 19 (1975), 269–271. (1975) MR0396344
  17. Le problème des régions voisines sur les surfaces closes orientables, J. Comb. Theory Ser. B 6 (1969), 177–195. (1969) Zbl0182.26601MR0234863
  18. Map Color Theorem, Springer, Berlin, 1974. (1974) Zbl0287.05102MR0349461
  19. Solution of the Heawood map-coloring problem, Proc. Natl. Acad. Sci. USA 60 (1968), 438–445. (1968) MR0228378
  20. The four color theorem, J. Combin. Theory Ser. B 70 (1997), 2–44. (1997) MR1441258
  21. An improved algorithm for testing the planarity of a graph, IEEE Trans. Comput. c-24 (1975), 113–121. (1975) Zbl0297.68030MR0414407
  22. Tilings, Handbook of Convex Geometry (Gruber P. M. , Will, J. M., eds.), Vol. B, North-Holland, Amsterdam, 1993. (1993) MR1242973
  23. Three-colour parallel multilevel preconditioner, Syst. Anal. Modelling Simulation 24 (1996), 255–261. (1996) Zbl0935.65045
  24. Graph Theory, Addison-Wesley, London, 1984. (1984) Zbl0554.05001MR0746795

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