Minimum 2-distance coloring of planar graphs and channel assignment

Article
First Online:
  • 32 Downloads

Abstract

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).

Keywords

Planar graph 2-Distance coloring Maximum degree 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The research work was supported by NFSC 11771403.

References

  1. 1.
    Agnarsson G, Halldorsson MM (2003) Coloring powers of planar graphs. SIAM J Discret Math 16:651–662MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bondy JA, Murty USR (2008) Graph Theory. Springer, New YorkCrossRefMATHGoogle Scholar
  3. 3.
    Borodin OV, Broersma HJ, Glebov A, Heuvel JVD (2002) Stars and bunches in planar graphs. Part II: general planar graphs and colourings, CDAM researches report 2002-05,Google Scholar
  4. 4.
    Zhu XB (2012) L(1,1)-Labeling of graphs. Zhejiang Normal UniversityGoogle Scholar
  5. 5.
    Yuehua Bu, Zhu Hongguo (2017) Strong edge-coloring of cubic planar graphs. Discret Math Algorithms Appl 9(1):1750013CrossRefMATHGoogle Scholar
  6. 6.
    Griggs JR, Yeh RK (1992) Labelling graphs with a condition at distance 2. SIAM J Discret Math 5:586C595MathSciNetCrossRefGoogle Scholar
  7. 7.
    van den Heuvel J, McGuinness SM (2003) Coloring of the square of planar graph. J Graph Theory 42:110–124MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jin J, Wei Y (2017) A note on 3-choosability of planar graphs under distance restrictions. Discret Math Algorithms Appl 9(1):1750011MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jonas TK (1993) Graph coloring analogues with a condition at distance two: L(2,1)-labellins and list \(\lambda \)-labellings, Ph.D. Thesis, University of South Carolina,Google Scholar
  10. 10.
    Karst N, Langowitz J, Oehrlein J, Troxell DS (2017) Radio \(k\)-chromatic number of cycles for large \(k\). Discret Math Algorithms Appl 9(3):1750031MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lisna PC, Sunitha MS (2015) \(b\)-Chromatic sum of a graph. Discret Math Algorithms Appl 7(4):1550040MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Molloy M, Salavatipour M (2005) A bound on the chromatic number of the square of a planar graph. J Combin Theory Ser B. 94:189–213MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Montassier M, Raspaud A (2006) A note on 2-facial coloring of plane graphs. Inf Process Lett 98:235–241MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pleanmani N, Panma S (2016) Bounds for the dichromatic number of a generalized lexicographic product of digraphs. Discret Math Algorithms Appl 8(2):1650034MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wegner G (1977) Graphs with given diameter and a coloring problem. Technical Report, University of Dortmund,Google Scholar
  16. 16.
    Thomasse C (2001) Applications of Tutte cycles, Technical report, Technical University of Denmark,Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingChina
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  3. 3.Zhejiang Normal University Xingzhi CollegeJinhuaChina

Personalised recommendations