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On the Density of C7-Critical Graphs

Abstract

In 1959, Grötzsch [5] famously proved that every planar graph of girth at least 4 is 3-colourable (or equivalently, admits a homomorphism to C3). A natural generalization of this is the following conjecture: for every positive integer t, every planar graph of girth at least 4t admits a homomorphism to C2t+1. This is in fact the planar dual of a well-known conjecture of Jaeger [7] which states that every 4t-edge-connected graph admits a modulo (2t + 1)-orientation. Though Jaeger’s original conjecture was disproved in [6], Lovász et al. [10] showed that every 6t-edge connected graph admits a modolo (2t + 1)-flow. The latter result implies that every planar graph of girth at least 6t admits a homomorphism to C2t+1. We improve upon this in the t = 3 case, by showing that every planar graph of girth at least 16 admits a homomorphism to C7. We obtain this through a more general result regarding the density of C7-critical graphs: if G is a C7-critical graph with G ∉ {C3, C5}, then \(e(G) \ge {{17v(G) - 2} \over {15}}\).

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References

  1. O. Borodin, S.-J. Kim, A. Kostochka and D. West: Homomorphisms from sparse graphs with large girth, Journal of Combinatorial Theory, Series B 90 (2004), 147–159.

    MathSciNet  Article  Google Scholar 

  2. D. W. Cranston and J. Li: Circular Flows in Planar Graphs, SIAM Journal on Discrete Mathematics 34 (2020), 497–519.

    MathSciNet  Article  Google Scholar 

  3. G. A. Dirac: Note on the colouring of graphs, Mathematische Zeitschrift 54 (1951), 347–353.

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. H. Grötzsch: Ein dreifarbensatz fur dreikreisfreie netze auf der kugel, Wiss. Z. Martin Luther Univ. Halle-Wittenberg, Math. Nat. Reihe 8 (1959), 109–120.

    MathSciNet  Google Scholar 

  6. M. Han, J. Li, Y. Wu and C. Zhang: Counterexamples to Jaeger’s circular flow conjecture, Journal of Combinatorial Theory, Series B 131 (2018), 1–11.

    MathSciNet  Article  Google Scholar 

  7. F. Jaeger: On circular flows in graphs, in: Finite and infinite sets, 391–402. Elsevier, 1984.

  8. W. Klostermeyer and C. Zhang: (2 + )-coloring of planar graphs with large odd-girth, Journal of Graph Theory 33 (2000), 109–119.

    MathSciNet  Article  Google Scholar 

  9. A. Kostochka and M. Yancey: Ore’s conjecture on color-critical graphs is almost true, Journal of Combinatorial Theory, Series B 109 (2014), 73–101.

    MathSciNet  Article  Google Scholar 

  10. L. Lovász, C. Thomassen, Y. Wu and C. Zhang: Nowhere-zero 3-flows and modulo k-orientations, Journal of Combinatorial Theory, Series B 103 (2013), 587–598.

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  12. O. Ore: The Four-Color Problem, Academic Press, New York, 1967.

    MATH  Google Scholar 

  13. X. Zhu: Circular chromatic number: a survey, Discrete Mathematics 229 (2001), 371–410.

    MathSciNet  Article  Google Scholar 

  14. X. Zhu: Circular chromatic number of planar graphs of large odd girth, The Electronic Journal of Combinatorics 8 (2001), 25.

    MathSciNet  Article  Google Scholar 

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Correspondence to Evelyne Smith-Roberge.

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Postle, L., Smith-Roberge, E. On the Density of C7-Critical Graphs. Combinatorica 42, 253–300 (2022). https://doi.org/10.1007/s00493-020-4177-y

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  • DOI: https://doi.org/10.1007/s00493-020-4177-y

Mathematics Subject Classification (2010)

  • 05C60