Skip to main content
SpringerLink
Log in

Fractional Coloring Planar Graphs under Steinberg-type Conditions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A Steinberg-type conjecture on circular coloring is recently proposed that for any prime p ≥ 5, every planar graph of girth p without cycles of length from p + 1 to p(p − 2) is Cp-colorable (that is, it admits a homomorphism to the odd cycle Cp). The assumption of p ≥ 5 being prime number is necessary, and this conjecture implies a special case of Jaeger’s Conjecture that every planar graph of girth 2p — 2 is Cp-colorable for prime p ≥ 5. In this paper, combining our previous results, we show the fractional coloring version of this conjecture is true. Particularly, the p = 5 case of our fractional coloring result shows that every planar graph of girth 5 without cycles of length from 6 to 15 admits a homomorphism to the Petersen graph.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Borodin, O. V., Glebov, A. N., Raspaud, A., et al.: Planar graphs without cycles of length from 4 to 7 are 3-colorable. J. Combin. Theory Ser. B, 93, 303–311 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borodin, O. V., Kim, S.-J., Kostochka, A. V., et al.: Homomorphisms from sparse graphs with large girth. J. Combin. Theory Ser. B, 90, 147–159 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen-Addad, V., Hebdige, M., Král’, D., et al.: Steinberg’s conjecture is false. J. Combin. Theory Ser. B, 122, 452–456 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cranston, D., Li, J.: Circular flows in planar graphs. SIAM J. Discrete Math., 34, 497–519 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dvořák, Z., Postle, L.: Density of 5/2-critical graphs. Combinatorica, 37, 863–886 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dvořák, Z., Škrekovski, R., Valla, T.: Planar graphs of odd-girth at least 9 are homomorphic to the Petersen graph. SIAM J. Discrete Math., 22, 568–591 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Galluccio, A., Goddyn, L., Hell, P.: High girth graphs avoiding a minor are nearly bipartite. J. Combin. Theory Ser. B, 83, 1–14 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. Han, M., Li, J., Wu, Y., et al.: Counterexamples to Jaeger’s Circular Flow Conjecture. J. Combin. Theory Ser. B, 131, 1–11 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hilton, A., Rado, R., Scott, S.: A (< 5)-color theorem for planar graphs. Bull. Lond. Math. Soc., 5, 302–306 (1973)

    Article  MATH  Google Scholar 

  11. Hu, X., Li, J.: Circular coloring and fractional coloring in planar graphs. J. Graph Theory, 2021 https://doi.org/10.1002/jgt.22742

  12. Jaeger, F.: Nowhere-zero flow problems, In: Selected Topics in Graph Theory 3, (L. W. Beineke and R. J. Wilson eds.), Academic Press, London, 71–95 (1988)

  13. Klostermeyer, W., Zhang, C.-Q.: (2 + ϵ)-coloring of planar graphs with large odd girth, J. Graph Theory, 33, 109–119 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lovász, L. M., Thomassen, C., Wu, Y., et al.: Nowhere-zero 3-flows and modulo k-orientations. J. Combin. Theory Ser. B, 103, 587–598 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Naserasr, R.: Mapping planar graphs into projective cubes. J. Graph Theory, 74(3), 249–259 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nesetril, J., Zhu, X.: On bounded tree-width duality of graphs. J. Graph Theory, 23, 151–162 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Postle, L., Smith-Roberge, E.: On the density of C7-critical graphs. Acta Math. Univ. Comenian. (N.S.), 88(3), 1009–1016 (2019)

    MathSciNet  Google Scholar 

  18. Thomassen, C.: The weak 3-flow conjecture and the weak circular flow conjecture. J. Combin. Theory Ser. B, 102, 521–529 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tutte, W. T.: A contribution on the theory of chromatic polynomial. Canad. J. Math., 6, 80–91 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhu, X.: Circular chromatic number of planar graphs of large odd girth. Electron. J. Combin., 8, R25, 11 pp. (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We sincerely thank all the referees for their time and helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P. R. China

    Xiao Lan Hu

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Jia Ao Li

Authors
  1. Xiao Lan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jia Ao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jia Ao Li.

Additional information

The first author was partially supported by the National Natural Science Foundation of China (Grant No. 11971196) and Hubei Provincial Science and Technology Innovation Base (Platform) Special Project 2020DFH002; the second author was partially supported by the National Natural Science Foundation of China (Grant Nos. 11901318, 12131013) and the Young Elite Scientists Sponsorship Program by Tianjin (Grant No. TJSQNTJ-2020-09)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, X.L., Li, J.A. Fractional Coloring Planar Graphs under Steinberg-type Conditions. Acta. Math. Sin.-English Ser. 39, 904–922 (2023). https://doi.org/10.1007/s10114-022-1085-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-1085-7

Keywords

MR(2010) Subject Classification

Search

Navigation