Edge lower bounds for list critical graphs, via discharging



A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon–Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.

Mathematics Subject Classification (2000)



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  1. [1]
    N. Alon and M. Tarsi: Colorings and orientations of graphs, Combinatorica 12 (1992), 125–134.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R. L. Brooks: On colouring the nodes of a network, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 37, Cambridge Univ Press, 1941, 194–197.Google Scholar
  3. [3]
    D. W. Cranston and D. B. West: An introduction to the discharging method via graph coloring, Discrete Math 340 (2017), 766–793. extended version: A guide to the discharging method, https://arxiv.org/abs/1306.4434v1 (2013).MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    G. A. Dirac: A theorem of R. L. Brooks and a conjecture of H. Hadwiger, Proceedings of the London Mathematical Society 3 (1957), no. 1, 161–195.Google Scholar
  5. [5]
    P. Erdős, A. L. Rubin and H. Taylor: Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. 26, 1979, 125–157.Google Scholar
  6. [6]
    T. Gallai: Kritische Graphen I., Publications of the Mathematical Institute of the Hungarian Academy of Sciences 8 (1963), 165–192 (in German).MathSciNetMATHGoogle Scholar
  7. [7]
    J. Hladký, D. Král’ and U. Schauz: Brooks’ theorem via the Alon–Tarsi theorem, Discrete Mathematics 310 (2010), 3426–3428.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    H. A. Kierstead and A. V. Kostochka: Ore-type versions of Brooks’ theorem, Journal of Combinatorial Theory, Series B 99 (2009), 298–305.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    H. A. Kierstead and L. Rabern: Improved lower bounds on the number of edges in list critical and online list critical graphs, arXiv preprint http://arxiv.org/abs/1406.7355 (2014).Google Scholar
  10. [10]
    A. V. Kostochka, L. Rabern and M. Stiebitz: Graphs with chromatic number close to maximum degree, Discrete Mathematics 312 (2012), 1273–1281.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    A. V. Kostochka and M. Stiebitz: A new lower bound on the number of edges in colour-critical graphs and hypergraphs, Journal of Combinatorial Theory, Series B 87 (2003), 374–402.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    A. V. Kostochka and M. Yancey: Ore’s conjecture for k=4 and Grötzsch’s theorem, Combinatorica 34 (2014), 323–329.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    A. V. Kostochka and M. Yancey: Ore’s conjecture on color-critical graphs is almost true, Journal of Combinatorial Theory, Series B 109 (2014), 73–101.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    M. Krivelevich: On the minimal number of edges in color-critical graphs, Combinatorica 17 (1997), 401–426.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    L. Rabern: Δ-critical graphs with small high vertex cliques, Journal of Combinatorial Theory, Series B 102 (2012), 126–130.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Riasat and U. Schauz: Critically paintable, choosable or colorable graphs, Discrete Mathematics 312 (2012), 3373–3383.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    U. Schauz: Mr. Paint and Mrs. Correct, The Electronic Journal of Combinatorics 16 (2009), R77.MathSciNetMATHGoogle Scholar
  18. [18]
    U. Schauz: Flexible color lists in Alon and Tarsi’s theorem, and time scheduling with unreliable participants, The Electronic Journal of Combinatorics 17 (2010), R13.MathSciNetMATHGoogle Scholar
  19. [19]
    M. Stiebitz: Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs, Combinatorica 2 (1982), 315–323.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    V. G. Vizing; Vextex coloring with given colors, Metody Diskretnogo Analiza 29 (1976), 3–10 (in Russian).Google Scholar
  21. [21]
    X. Zhu: On-line list colouring of graphs, The Electronic Journal of Combinatorics 16 (2009), R127.MathSciNetMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  2. 2.LBD Data SolutionsLancasterUSA

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