Large faces in 4-critical planar graphs with minimum degree 4


We prove that the size of the largest face of a 4-critical planar graph with δ≥4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows\(\mathop {\lim }\limits_{n \to \infty } \frac{{f(n)}}{n} = \frac{1}{2}\).

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  1. [1]

    H. L. Abbott, M. Katchalski, andB. Zhou: Proof of a conjecture of Dirac concerning 4-critical planar graphs.Discrete Math.,132 (1994), 367–371.

    Google Scholar 

  2. [2]

    H. L. Abbott andB. Zhou: The edge density of 4-critical planar graphs.Combinatorica,11 (1991), 185–189.

    Google Scholar 

  3. [3]

    T. Gallai: Critical graphs. InTheory of Graphs and its Applications (Proc. Symp. Smolenice, 1963), 43–45, Publ. House Szech. Acad. Sci., 1964.

  4. [4]

    B. Grünbaum: The edge density of 4-critical planar graphs.Combinatorica,8 (1988), 137–139.

    Google Scholar 

  5. [5]

    G. Koester: Note to a problem of T. Gallai and G. A. Dirac.Combinatorica,5 (1985), 227–228.

    Google Scholar 

  6. [6]

    G. Koester: 4-critical 4-valent planar graphs constructed with crowns.Math. Scand.,67 (1990), 17–22.

    Google Scholar 

  7. [7]

    G. Koester: On 4-critical planar graphs with high edge density.Discrete Math.,98 (1991), 147–151.

    Google Scholar 

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All three authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada.

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Abbott, H.L., Hare, D.R. & Zhou, B. Large faces in 4-critical planar graphs with minimum degree 4. Combinatorica 15, 455–467 (1995).

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Mathematics Subject Classification (1991)

  • Primary 05C15
  • Secondary 05C35