, Volume 35, Issue 5, pp 619–631 | Cite as

The edge density of critical digraphs

Original Paper


Let χ(G) denote the chromatic number of a graph G. We say that G is k-critical if χ(G)=k and χ(H) < k for every proper subgraph HG. Over the years, many properties of k-critical graphs have been discovered, including improved upper and lower bounds for ||G||, the number of edges in a k-critical graph, as a function of |G|, the number of vertices. In this note, we analyze this edge density problem for directed graphs, where the chromatic number χ(D) of a digraph D is defined to be the fewest number of colours needed to colour the vertices of D so that each colour class induces an acyclic subgraph. For each k ≥ 3, we construct an infinite family of sparse k-critical digraphs for which \(\left\| D \right\| < \left( {\tfrac{{k^2 - k + 1}} {2}} \right)\left| D \right|\) and an infinite family of dense k-critical digraphs for which \(\left\| D \right\| > \left( {\tfrac{1} {2} - \tfrac{1} {{2^k - 1}}} \right)\left| D \right|^2\). One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair of integers k,l≥3. This extends a result by Bokal et al. who used a probabilistic approach to demonstrate the existence of one such digraph.

Mathematics Subject Classification (2000)

05C20 05C35 05C40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Araujo-Pardo and M. Olsen: A conjecture of Neumann-Lara on infinite families of r-dichromatic circulant tournaments, Discrete Mathematics 310 (2010), 489–492.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    E. Berger, K. Choromanski, M. Chudnovsky, J. Fox, M. Loebl, A. Scott, P. Seymour and S. Thomassé: Tournaments and colouring, Journal of Combinatorial Theory Ser. B 103 (2013), 1–20.MATHCrossRefGoogle Scholar
  3. [3]
    D. Bokal, G. Fijavz, M. Juvan, P. Kayll and B. Mohar: The circular chromatic number of a digraph, Journal of Graph Theory 46 (2004), 227–240.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    B. Bollobás: Chromatic number, girth, and maximal degree, Discrete Mathematics 24 (1978), 311–314.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    G. A. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, Journal of the London Mathematical Society 27 (1952), 85–92.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    P. Erdős and A. Hajnal: Ramsey-type theorems, Discrete Applied Mathematics 25 (1989), 37–52.MathSciNetCrossRefGoogle Scholar
  7. [7]
    B. Farzad and M. Molloy: On the edge density of 4-critical graphs, Combinatorica 29 (2009), 665–689.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    T. Gallai: Kritische Graphen I, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 165–192.MATHMathSciNetGoogle Scholar
  9. [9]
    A. Harutyunyan and B. Mohar: Gallai’s theorem for list coloring of digraphs, SIAM Journal on Discrete Mathematics 25 (2011), 170–180.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Harutyunyan and B. Mohar: Two results on the digraph chromatic number, Discrete Mathematics 312 (2012), 1823–1826.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    T. R. Jensen: Dense critical and vertex-critical graphs, Discrete Mathematics 258 (2002), 63–84.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    T. R. Jensen and B. Toft: Graph coloring problems, New York, Wiley Interscience, 1995.MATHGoogle Scholar
  13. [13]
    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.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. V. Kostochka and M. Yancey: Ore’s Conjecture on color-critical graphs is almost true, to appear.Google Scholar
  15. [15]
    V. Neumann-Lara: The dichromatic number of a digraph, Journal of Combinatorial Theory Series B 33 (1982), 265–270.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    V. Neumann-Lara: The 3 and 4-dichromatic tournaments of minimum order, Discrete Mathematics 135 (1994), 233–243.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    B. Mohar: Circular colorings of edge-weighted graphs, Journal of Graph Theory 43 (2003), 107–116.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    B. Mohar: Eigenvalues and colorings of graphs, Linear Algebra and its Applications 432 (2010), 2273–2277.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    B. Toft: On the maximal number of edges of critical k-chromatic graphs, Studia Scientiarum Mathematicarum Hungarica 5 (1970), 461–470.MathSciNetGoogle Scholar
  20. [20]
    P. Turán: On an extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941), 436–452.MathSciNetGoogle Scholar
  21. [21]
    H. S. Wilf: The eigenvalues of a graph and its chromatic number, Journal of the London Mathematical Society 42 (1967), 330–332.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Quest University CanadaSquamishCanada
  2. 2.JST-ERATO Kawarabayashi Large Graph ProjectNational Institute of InformaticsTokyoJapan

Personalised recommendations