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2012 | 32 | 1 | 141-151
Tytuł artykułu

2-distance 4-colorability of planar subcubic graphs with girth at least 22

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.
Słowa kluczowe
Wydawca
Rocznik
Tom
32
Numer
1
Strony
141-151
Opis fizyczny
Daty
wydano
2012
otrzymano
2010-11-23
poprawiono
2011-03-24
zaakceptowano
2011-03-26
Twórcy
  • Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, and Novosibirsk State University, Novosibirsk, 630090, Russia
  • Institute of Mathematics at Yakutsk State University and North-Eastern Federal University, Yakutsk, 677891, Russia
Bibliografia
  • [1] G. Agnarsson and M.M. Halldorsson, Coloring powers of planar graphs, in: Combinatorics, Proc. SODA'00 (SIAM, 2000) 654-662.
  • [2] G. Agnarsson and M.M. Halldorsson, Coloring powers of planar graphs, SIAM J. Discrete Math. 16 (2003) 651-662, doi: 10.1137/S0895480100367950.
  • [3] O.V. Borodin, H.J. Broersma, A.N. Glebov and J. van den Heuvel, The minimum degree and chromatic number of the square of a planar graph, Diskretn. Anal. Issled. Oper., 8 no. 4 (2001) 9-33 (in Russian).
  • [4] O.V. Borodin, H.J. Broersma, A.N. Glebov and J. van den Heuvel, The structure of plane triangulations in terms of stars and bunches, Diskretn. Anal. Issled. Oper. 8 no. 2 (2001) 15-39 (in Russian).
  • [5] O.V. Borodin, A.N. Glebov, A.O. Ivanova, T.K. Neustroeva and V.A. Tashkinov, Sufficient conditions for the 2-distance Δ+1-colorability of plane graphs, Sib. Elektron. Mat. Izv. 1 (2004) 129-141 (in Russian).
  • [6] O.V. Borodin and A.O. Ivanova, 2-distance (Δ+2)-coloring of planar graphs with girth six and Δ≥ 18, Discrete Math. 309 (2009) 6496-6502, doi: 10.1016/j.disc.2009.06.029.
  • [7] O.V. Borodin and A.O. Ivanova, List 2-distance (Δ+2)-coloring of planar graphs with girth six, Europ. J. Combin. 30 (2009) 1257-1262, doi: 10.1016/j.ejc.2008.12.019.
  • [8] O.V. Borodin and A.O. Ivanova, 2-distance 4-coloring of planar subcubic graphs, Diskretn. Anal. Issled. Oper. 18 no. 2 (2011) 18-28 (in Russian).
  • [9] O.V. Borodin, A.O. Ivanova and T.K. Neustroeva, 2-distance coloring of sparse plane graphs, Sib. Elektron. Mat. Izv. 1 (2004) 76-90 (in Russian).
  • [10] O.V. Borodin, A.O. Ivanova and T.K. Neustroeva, Sufficient conditions for 2-distance (Δ+1)-colorability of planar graphs of girth 6, Diskretn. Anal. Issled. Oper. 12 no.3 (2005) 32-47 (in Russian).
  • [11] O.V. Borodin, A.O. Ivanova and T.K. Neustroeva, Sufficient conditions for the minimum 2-distance colorability of planar graphs with girth 6, Sib. Elektron. Mat. Izv. 3 (2006) 441-450 (in Russian).
  • [12] Z. Dvořák, D. Kràl, P. Nejedlỳ and R. Škrekovski, Coloring squares of planar graphs with girth six, European J. Combin. 29 (2008) 838-849, doi: 10.1016/j.ejc.2007.11.005.
  • [13] Z. Dvořák, R. Škrekovski and M. Tancer, List-coloring squares of sparse subcubic graphs, SIAM J. Discrete Math. 22 (2008) 139-159, doi: 10.1137/050634049.
  • [14] F. Havet, Choosability of the square of planar subcubic graphs with large girth, Discrete Math. 309 (2009) 3353-3563, doi: 10.1016/j.disc.2007.12.100.
  • [15] A.O. Ivanova, List 2-distance (Δ+1)-coloring of planar graphs with girth at least 7, Diskretn. Anal. Issled. Oper. 17 no.5 (2010) 22-36 (in Russian).
  • [16] A.O. Ivanova and A.S. Solov'eva, 2-Distance (Δ +2)-coloring of sparse planar graphs with Δ=3, Mathematical Notes of Yakutsk University 16 no. 2 (2009) 32-41 (in Russian).
  • [17] T. Jensen and B. Toft, Graph Coloring Problems (New York: John Willey & Sons, 1995).
  • [18] M. Molloy and M.R. Salavatipour, Frequency channel assignment on planar networks, in: LNCS, ed(s), R.H. Möhring and R. Raman (Springer, 2002) 736-747.
  • [19] M. Molloy and M.R. Salavatipour, A bound on the 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.
  • [20] M. Montassier and A. Raspaud, A note on 2-facial coloring of plane graphs, Inform. Process. Lett. 98 (2006) 235-241, doi: 10.1016/j.ipl.2006.02.013.
  • [21] G. Wegner, Graphs with Given Diameter and a Coloring Problem (Technical Report, University of Dortmund, Germany, 1977).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1592
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