PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2009 | 29 | 3 | 521-543
Tytuł artykułu

Colouring vertices of plane graphs under restrictions given by faces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions.
Wydawca
Rocznik
Tom
29
Numer
3
Strony
521-543
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-03-28
poprawiono
2008-08-19
zaakceptowano
2008-09-05
Twórcy
autor
  • Institute of Mathematics, P.J. Šafárik University, Jesenná 5, SK-04001 Košice, Slovakia
  • Institute of Mathematics, P.J. Šafárik University, Jesenná 5, SK-04001 Košice, Slovakia
Bibliografia
  • [1] K. Appel and W. Haken, Every planar map is four colorable, Contemporary Mathematics 98 (American Mathematical Society, 1989).
  • [2] K. Budajová, S. Jendrol and S. Krajci, Parity vertex colouring of graphs, manuscript (2007).
  • [3] D.P. Bunde, K. Milans, D.B. West and H. Wu, Parity and strong parity edge-coloring of graphs, manuscript (2006).
  • [4] G. Chartrand and L. Lesniak, Graphs and Digraphs (Chapman & HALL/CRC, Boca Raton, 2005).
  • [5] H. Enomoto, M. Hornák and S. Jendrol, Cyclic chromatic number of 3-connected plane graphs, SIAM J. Discrete Math. 14 (2001) 121-137, doi: 10.1137/S0895480198346150.
  • [6] M. Hornák and S. Jendrol, On a conjecture by Plummer and Toft, J. Graph Theory 30 (1999) 177-189, doi: 10.1002/(SICI)1097-0118(199903)30:3<177::AID-JGT3>3.0.CO;2-K
  • [7] M. Hornák and J. Zlámalová, Another step towards proving a conjecture by Plummer and Toft, IM Preprint, series A, No.11/2006 (2006).
  • [8] S. Jendrol, Rainbowness of cubic plane graphs, Discrete Math. 306 (2006) 3321-3326, doi: 10.1016/j.disc.2006.06.012.
  • [9] V. Jungic, D. Král and R. Skrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2006) 169-182, doi: 10.1007/s00493-006-0012-3.
  • [10] H. Lebesgue, Quelques consequences simple de la formula d'Euler, J. de Math. Pures Appl. 9 (1940) 27-43.
  • [11] M. Molloy and M.R. Salavatipour, A bound on the cyclic chromatic number of the square of a planar graph, J. Combin. Theory (B) 94 (2005) 189-213, doi: 10.1016/j.jctb.2004.12.005.
  • [12] O. Ore and M.D. Plummer, Cyclic coloration of plane graphs, in: W.T. Tutte, Recent Progress in Combinatorics Academic Press (1969) 287-293.
  • [13] M.D. Plummer and B. Toft, Cyclic coloration of 3-polytopes, J. Graph Theory 11 (1987) 507-515, doi: 10.1002/jgt.3190110407.
  • [14] N. Rampersad, A note on non-repetitive colourings of planar graphs, manuscript (2003).
  • [15] R. Ramamurthi and D.B. West, Maximum face-constrained coloring of plane graphs, Discrete Math. 274 (2004) 233-240, doi: 10.1016/j.disc.2003.09.001.
  • [16] D.P. Sanders and Y. Zhao, A new bound on the cyclic chromatic number, J. Combin. Theory (B) 83 (2001) 102-111, doi: 10.1006/jctb.2001.2046.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1462
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.