Skip to main content
SpringerLink
Log in

3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle

  • Conference paper
  • First Online:
Combinatorial Algorithms (IWOCA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8986))

Included in the following conference series:

  • International Workshop on Combinatorial Algorithms

  • 673 Accesses

Abstract

Given a triangle-free planar graph G and a cycle C of length 9 in G, we characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results for the length of C up to 8.

Ilkyoo Choi—Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653).

Jan Ekstein—Supported by P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund (ERDF), project NTIS - New Technologies for the Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

Přemysl Holub—Supported by NSF grants DMS-1266016.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aksenov, V.A.: The extension of a \(3\)-coloring on planar graphs. Diskret. Analiz (Vyp. 26 Grafy i Testy), 3–19, 84 (1974)

    Google Scholar 

  2. Aksenov, V.A., Borodin, O.V., Glebov, A.N.: Continuation of a 3-coloring from a 6-face to a plane graph without 3-cycles. Diskretn. Anal. Issled. Oper. Ser. 1 10(3), 3–11 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Aksenov, V.A., Borodin, O.V., Glebov, A.N.: Continuation of a 3-coloring from a 7-face onto a plane graph without 3-cycles. Sib. Èlektron. Mat. Izv. 1, 117–128 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Appel, K., Haken, W.: Every planar map is four colorable. I. Discharging. Illinois J. Math. 21(3), 429–490 (1977)

    MATH  MathSciNet  Google Scholar 

  5. Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. II. Reducibility. Illinois J. Math. 21(3), 491–567 (1977)

    MATH  MathSciNet  Google Scholar 

  6. Borodin, O.V.: A new proof of Grünbaum’s \(3\) color theorem. Discrete Math. 169(1–3), 177–183 (1997). http://dx.doi.org/10.1016/0012-365X(95)00984-5

    Article  MATH  MathSciNet  Google Scholar 

  7. Borodin, O.V.: Colorings of plane graphs: A survey. Discrete Math. 33(4), 517–539 (2013). http://dx.doi.org/10.1016/j.disc.2012.11.011

    Article  Google Scholar 

  8. Borodin, O.V., Dvořák, Z., Kostochka, A.V., Lidický, B., Yancey, M.: Planar 4-critical graphs with four triangles. Eur. J. Combin. 41, 138–151 (2014). http://dx.doi.org/10.1016/j.ejc.2014.03.009

    Article  MATH  Google Scholar 

  9. Dailey, D.P.: Uniqueness of colorability and colorability of planar \(4\)-regular graphs are NP-complete. Discrete Math. 30(3), 289–293 (1980). http://dx.doi.org/10.1016/0012-365X(80)90236-8

    Article  MATH  MathSciNet  Google Scholar 

  10. Dvořák, Z., Lidický, B.: 4-critical graphs on surfaces without contractible \((\le 4)\)-cycles. SIAM J. Discrete Math. 28(1), 521–552 (2014). http://dx.doi.org/10.1137/130920952

    Article  MATH  MathSciNet  Google Scholar 

  11. Dvořák, Z., Kawarabayashi, K.i.: Choosability of planar graphs of girth 5. ArXiv e-prints, September 2011

    Google Scholar 

  12. Dvořák, Z., Král, D., Thomas, R.: Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle (2013) (Submitted)

    Google Scholar 

  13. Dvořák, Z., Král, D., Thomas, R.: Three-coloring triangle-free graphs on surfaces IV. \(4\)-faces in critical graphs (2014) (Manuscript)

    Google Scholar 

  14. Dvořák, Z., Lidický, B.: 3-coloring triangle-free planar graphs with a precolored 8-cycle (2014). http://dx.doi.org/10.1002/jgt.21842 (Accepted to J. Graph Theory)

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  16. Gimbel, J., Thomassen, C.: Coloring graphs with fixed genus and girth. Trans. Amer. Math. Soc. 349(11), 4555–4564 (1997). http://dx.doi.org/10.1090/S0002-9947-97-01926-0

    Article  MATH  MathSciNet  Google Scholar 

  17. Grötzsch, H.: Ein Dreifarbenzatz für Dreikreisfreie Netze auf der Kugel. Math.Natur. Reihe 8, 109–120 (1959)

    Google Scholar 

  18. Grünbaum, B.: Grötzsch’s theorem on \(3\)-colorings. Michigan Math. J. 10, 303–310 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  19. Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Combin. Theory Ser. B 70(1), 2–44 (1997). http://dx.doi.org/10.1006/jctb.1997.1750

    Article  MATH  MathSciNet  Google Scholar 

  20. Thomassen, C.: The chromatic number of a graph of girth 5 on a fixed surface. J. Combin. Theory Ser. B 87(1), 38–71 (2003). http://dx.doi.org/10.1016/S0095-8956(02)00027-8 (dedicated to Crispin St. J. A. Nash-Williams)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tutte, W.T.: A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80–91 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  22. Walls, B.H.: Coloring girth restricted graphs on surfaces. ProQuest LLC, Ann Arbor (1999). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9953838 (thesis (Ph.D.)–Georgia Institute of Technology)

Download references

Author information

Authors and Affiliations

  1. Korea Advanced Institute of Science and Technology, Daejeon, South Korea

    Ilkyoo Choi

  2. University of West Bohemia, Pilsen, Czech Republic

    Jan Ekstein & Přemysl Holub

  3. Iowa State University, Ames, USA

    Bernard Lidický

Authors
  1. Ilkyoo Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Ekstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Přemysl Holub
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bernard Lidický
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard Lidický .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic

    Kratochvíl Jan

  2. University of Newcastle, Newcastle, New South Wales, Australia

    Mirka Miller

  3. University of Minnesota, Duluth Department of Mathematics, Duluth, Minnesota, USA

    Dalibor Froncek

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Choi, I., Ekstein, J., Holub, P., Lidický, B. (2015). 3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle. In: Jan, K., Miller, M., Froncek, D. (eds) Combinatorial Algorithms. IWOCA 2014. Lecture Notes in Computer Science(), vol 8986. Springer, Cham. https://doi.org/10.1007/978-3-319-19315-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19315-1_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19314-4

  • Online ISBN: 978-3-319-19315-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation