Graphs and Combinatorics

, Volume 29, Issue 5, pp 1489–1499 | Cite as

Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth

Original Paper
First Online:
Received:
Revised:

Abstract

We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a \({\overleftrightarrow{K_4}}\) -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here \({\overleftrightarrow{K_n}}\) means a digraph with n vertices and arcs in both directions between every distinct pair), and a \({\overleftrightarrow{K_3}}\) -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.

Keywords

Digraph Homomorphism Homomorphism bound Paley tournament 

Mathematics Subject Classification (2000)

Primary 05C15 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Marshall T.H.: Homomorphism bounds for oriented planar graphs. J. Graph Theory 55, 175–190 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Raspaud A., Sopena E.: Good and semi-strong colorings of oriented planar graphs. Inform. Proc. Lett. 51, 171–174 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Marshall, T.H.: On oriented graphs with certain extension properties. Ars Combinatoria (2012, in press)Google Scholar
  4. 4.
    Borodin, Ivanova, A.O.: An oriented 7-coloring of planar graphs with girth at least 7. Sib. Electron. Math. Rep. 2, 222–229 (2005)Google Scholar
  5. 5.
    Borodin O.V., Ivanova A.O., Kostochka A.V.: Oriented 5-coloring of vertices in sparse graphs. (Russian) Diskretn. Anal. Issled. Oper. Ser. 1 13(1), 16–32 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Borodin O.V., Kostochka A.V., Nešetřil J., Raspaud A., Sopena E.: On the maximum average degree and the oriented chromatic number of a graph. Discret. Math. 206, 77–90 (1999)MATHCrossRefGoogle Scholar
  7. 7.
    Nešetřil, J., Raspaud, A., Sopena, E.: Colorings and girth of oriented planar graphs. Discret. Math. 165–166, 519–530 (1997)Google Scholar
  8. 8.
    Ochem P.: Oriented colorings of triangle-free planar graphs. Inform. Process. Lett. 92, 71–76 (2004)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ochem P., Pinlou A.: Oriented colorings of partial 2-trees. Inform. Process. Lett. 108(2), 82–86 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ochem P., Pinlou A.: Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs. Elect. Notes Discret. Math. 37, 123–128 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pinlou A.: An oriented coloring of planar graphs with girth at least five. Discret. Math. 309, 2108–2118 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Thomassen C.: Every planar graph is 5-choosable. J. Combin. Theory Ser. B 62, 180–181 (1994)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Thomassen C.: A short list color proof of Grötzsch’s theorem. J. Combin. Theory Ser. B 88(1), 189–192 (2003)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Sopena E.: There exist oriented planar graphs with oriented chromatic number at least sixteen. Inform. Proc. Lett. 81, 309–312 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pinlou A., Sopena E.: Oriented vertex and arc colorings of outerplanar graphs. Inform. Proc. Lett. 100, 97–104 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Marshall, T.H.: Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth. KAM Dimatia, Czech Republic (2011, preprint)Google Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAmerican University of SharjahSharjahUnited Arab Emirates

Personalised recommendations