Opuscula Math. 38, no. 6 (2018), 819-827
https://doi.org/10.7494/OpMath.2018.38.6.819

 
Opuscula Mathematica

Zig-zag facial total-coloring of plane graphs

Július Czap
Stanislav Jendroľ
Margit Voigt

Abstract. In this paper we introduce the concept of zig-zag facial total-coloring of plane graphs. We obtain lower and upper bounds for the minimum number of colors which is necessary for such a coloring. Moreover, we give several sharpness examples and formulate some open problems.

Keywords: plane graph, facial coloring, total-coloring, zig-zag coloring.

Mathematics Subject Classification: 05C10, 05C15.

Full text (pdf)

  1. K. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), 711-712.
  2. J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, 2008.
  3. R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Philosophical Society, Math. Phys. Sci. 37 (1941), 194-197.
  4. J. Czap, S. Jendroľ, Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017), 2691-2703.
  5. J. Czap, P. Šugerek, Facial total-coloring of bipartite plane graphs, Int. J. Pure Appl. Math. 115 (2017), 115-121.
  6. I. Fabrici, S. Jendroľ, M. Vrbjarová, Facial entire colouring of plane graphs, Discrete Math. 339 (2016), 626-631.
  7. I. Fabrici, S. Jendroľ, M. Voigt, Facial list colourings of plane graphs, Discrete Math. 339 (2016), 2826-2831.
  8. H. Grötzsch, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin Luther-Universität, Halle, Wittenberg, Math.-Nat. Reihe 8 (1959), 109-120.
  9. S. Kitaev, Patterns in permutations and words, Springer, 2011.
  10. T.L. Saaty, P.C. Kainen, The Four-Color Problem, Assaults and Conquest, McGraw-Hill, 1977.
  11. P.G. Tait, Remarks on the colouring of maps, Proc. Roy. Soc. Edinburgh 10 (1880), 729.
  • Július Czap
  • Technical University of Košice, Department of Applied Mathematics and Business Informatics, Faculty of Economics, Němcovej 32, 04001 Košice, Slovakia
  • Stanislav Jendroľ
  • P.J. Šafárik University, Institute of Mathematics, Jesenná 5, 04001 Košice, Slovakia
  • Margit Voigt
  • University of Applied Sciences Dresden, Faculty of Informatics and Mathematics, Friedrich-List-Platz 1, 01069 Dresden, Germany
  • Communicated by Adam Paweł Wojda.
  • Received: 2018-01-24.
  • Revised: 2018-02-13.
  • Accepted: 2018-02-15.
  • Published online: 2018-07-05.
Opuscula Mathematica - cover

Cite this article as:
Július Czap, Stanislav Jendroľ, Margit Voigt, Zig-zag facial total-coloring of plane graphs, Opuscula Math. 38, no. 6 (2018), 819-827, https://doi.org/10.7494/OpMath.2018.38.6.819

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.