Graphs and Combinatorics

, Volume 33, Issue 4, pp 751–787 | Cite as

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

  • Zhanar Berikkyzy
  • Christopher Cox
  • Michael Dairyko
  • Kirsten Hogenson
  • Mohit Kumbhat
  • Bernard LidickýEmail author
  • Kacy Messerschmidt
  • Kevin Moss
  • Kathleen Nowak
  • Kevin F. Palmowski
  • Derrick Stolee
Original Paper


All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any \(\ell \in \{3,4,5,6,7\}\), a planar graph is 4-choosable if it is \(\ell \)-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded \(\ell \)-cycle is an \(\ell \)-cycle with one additional edge. We demonstrate for each \(\ell \in \{5,6,7\}\) that a planar graph is (4, 2)-choosable if it does not contain chorded \(\ell \)-cycles.


Graph coloring Planar graph Choosability with separation Discharging 



We are grateful to anonymous referee for spotting mistakes in the previous version of the manuscript. We thank Ryan R. Martin, Alex Nowak, Alex Schulte, and Shanise Walker for participation in the early stages of the Project.


  1. 1.
    Alon, N., Tarsi, M.: Colorings and Orientations of Graphs. Combinatorica 12(2), 125–134 (1992). doi: 10.1007/BF01204715 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borodin, O.V.: Colorings of Plane Graphs: A Survey. Discret. Math. 313(4), 517–539 (2013). doi: 10.1016/j.disc.2012.11.011 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, O.V., Ivanova, A.O.: Planar Graphs Without Triangular 4-Cycles are 4-Choosable. Sib. Élektron. Mat. Izv. 5, 75–79 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Choi, I., Lidický, B., Stolee, D.: On Choosability with Separation of Planar Graphs with Forbidden Cycles. J. Graph Theory 81(3), 283–306 (2016). doi: 10.1002/jgt.21875 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cranston, D.W., West, D.B.: An Introduction to the Discharging Method via Graph Coloring. Discret. Math. 340(4), 766–793 (2017). doi: 10.1016/j.disc.2016.11.022 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farzad, B.: Planar Graphs Without 7-Cycles are 4-Choosable. SIAM J. Discret. Math. 23(3), 1179–1199 (2009). doi: 10.1137/05064477X MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fijavž, G., Juvan, M., Mohar, B., Škrekovski, R.: Planar Graphs Without Cycles of Specific Lengths. Eur. J. Combin. 23(4), 377–388 (2002). doi: 10.1006/eujc.2002.0570 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kierstead, H.A., Lidický, B.: On Choosability with Separation of Planar Graphs with Lists of Different Sizes. Discrete Math. 338(10), 1779–1783 (2015). doi: 10.1016/j.disc.2015.01.008 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kratochvíl, J., Tuza, Z., Voigt, M.: Brooks-type Theorems for Choosability with Separation. J. Graph Theory 27(1), 43–49 (1998). doi: 10.1002/(SICI)1097-0118(199801)27:1<43::AID-JGT7>3.3.CO;2-Q
  10. 10.
    Lam, P.C.B., Xu, B., Liu, J.: The \(4\)-Choosability of Plane Graphs Without \(4\)-Cycles. J. Combin. Theory Ser. B 76(1), 117–126 (1999). doi: 10.1006/jctb.1998.1893 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Thomassen, C.: Every Planar Graph is \(5\)-Choosable. J. Combin. Theory Ser. B 62(1), 180–181 (1994). doi: 10.1006/jctb.1994.1062 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Voigt, M.: List Colourings of Planar Graphs. Discret. Math. 120(1–3), 215–219 (1993). doi: 10.1016/0012-365X(93)90579-I MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Škrekovski, R.: A Note on Choosability with Separation for Planar Graphs. Ars Combin. 58, 169–174 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Wang, W., Lih, K.W.: Choosability and Edge Choosability Of Planar Graphs Without Five Cycles. Appl. Math. Lett. 15(5), 561–565 (2002). doi: 10.1016/S0893-9659(02)80007-6 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)Google Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Zhanar Berikkyzy
    • 1
  • Christopher Cox
    • 2
  • Michael Dairyko
    • 1
  • Kirsten Hogenson
    • 3
  • Mohit Kumbhat
    • 4
  • Bernard Lidický
    • 1
    Email author
  • Kacy Messerschmidt
    • 1
  • Kevin Moss
    • 1
  • Kathleen Nowak
    • 5
  • Kevin F. Palmowski
    • 1
  • Derrick Stolee
    • 6
  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburgUSA
  3. 3.Department of Mathematics and Computer ScienceColorado CollegeColorado SpringsUSA
  4. 4.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA
  5. 5.Pacific Northwest National LaboratoryRichlandUSA
  6. 6.MicrosoftDurhamUSA

Personalised recommendations