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

Abstract

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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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). https://doi.org/10.1007/BF01192518

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Mathematics Subject Classification (1991)

  • Primary 05C15
  • Secondary 05C35