On the structural result on normal plane maps

• Autorzy: Tomás Madaras , Andrea Marcinová
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

plane map; distance colouring

Kategorie tematyczne

• 05C10: Planar graphs; geometric and topological aspects of graph theory

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2002
• Tom: 22
• Numer: 2
• Strony: 293-303

Daty

wydano

2002

otrzymano

2001-04-24

poprawiono

2002-03-23

Twórcy

autor

Tomás Madaras

• Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic

autor

Andrea Marcinová

• Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic

Bibliografia

• [1] P. Baldi, On a generalized family of colourings, Graphs and Combinatorics 6 (1990) 95-110, doi: 10.1007/BF01787722.
• [2] O. Borodin, Joint generalization of theorems by Lebesgue and Kotzig, Diskret. Mat. 3 (1991) 24-27 (in Russian).
• [3] O. Borodin, Joint colouring of vertices, edges and faces of plane graphs, Metody Diskret. Analiza 47 (1988) 27-37 (in Russian).
• [4] M. Fellows, P. Hell and K. Seyffarth, Constructions of dense planar networks, manuscript, 1993.
• [5] S. Jendrol' and Z. Skupień, On the vertex/edge distance colourings of planar graphs, Discrete Math. 236 (2001) 167-177.
• [6] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. Cas. SAV 5 (1955) 233-237 (in Slovak).
• [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.
• [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] H. Lebesgue, Quelques conséquences simples de la formule d'Euler, J. Math. Pures Appl. (9) 19 (1940) 27-43.
• [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] J. van den Heuvel and S. McGuinness, Colouring the Square of a Planar Graph, preprint, 2001.
• [12] G. Wegner, Graphs with given diameter and a coloring problem (Technical report, University of Dortmund, 1977).

Identyfikatory

DOI

10.7151/dmgt.1176