# On the structural result on normal plane maps

• Volume: 22, Issue: 2, page 293-303
• ISSN: 2083-5892

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## Abstract

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We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6+\left[\left(2D+12\right)/\left(D-2\right)\right]\left({\left(D-1\right)}^{\left(t-1\right)}-1\right)$. This improves a recent bound $6+\left[\left(3D+3\right)/\left(D-2\right)\right]\left({\left(D-1\right)}^{t-1}-1\right)$, D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.

## How to cite

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Tomás Madaras, and Andrea Marcinová. "On the structural result on normal plane maps." Discussiones Mathematicae Graph Theory 22.2 (2002): 293-303. <http://eudml.org/doc/270155>.

abstract = {We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6 + [(2D+12)/(D-2)]((D-1)^\{(t-1)\} - 1)$. This improves a recent bound $6 + [(3D+3)/(D-2)]((D-1)^\{t-1\}-1)$, D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.},
author = {Tomás Madaras, Andrea Marcinová},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {plane map; distance colouring},
language = {eng},
number = {2},
pages = {293-303},
title = {On the structural result on normal plane maps},
url = {http://eudml.org/doc/270155},
volume = {22},
year = {2002},
}

TY - JOUR
AU - Andrea Marcinová
TI - On the structural result on normal plane maps
JO - Discussiones Mathematicae Graph Theory
PY - 2002
VL - 22
IS - 2
SP - 293
EP - 303
AB - We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6 + [(2D+12)/(D-2)]((D-1)^{(t-1)} - 1)$. This improves a recent bound $6 + [(3D+3)/(D-2)]((D-1)^{t-1}-1)$, D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.
LA - eng
KW - plane map; distance colouring
UR - http://eudml.org/doc/270155
ER -

## References

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6. [6] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. Cas. SAV 5 (1955) 233-237 (in Slovak).
7. [7] F. Kramer and H. Kramer, Un problème de coloration des sommets d'un graphe, C.R. Acad. Sci. Paris Sér. A-B 268 (1969) A46-A48. Zbl0165.57302
8. [8] F. Kramer and H. Kramer, On the generalized chromatic number, in: Combinatorics '84 (Bari, Italy) North-Holland Math. Stud. 123 (North-Holland, Amsterdam-New York, 1986) (and Annals Discrete Math. 30, 1986) 275-284.
9. [9] H. Lebesgue, Quelques conséquences simples de la formule d'Euler, J. Math. Pures Appl. (9) 19 (1940) 27-43. Zbl0024.28701
10. [10] Z. Skupień, Some maximum multigraphs and edge/vertex distance colourings, Discuss. Math. Graph Theory 15 (1995) 89-106, doi: 10.7151/dmgt.1010.
11. [11] J. van den Heuvel and S. McGuinness, Colouring the Square of a Planar Graph, preprint, 2001. Zbl1008.05065
12. [12] G. Wegner, Graphs with given diameter and a coloring problem (Technical report, University of Dortmund, 1977).

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