Ore’s conjecture for k=4 and Grötzsch’s Theorem

Abstract

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k−1)-colorable. Let f k (n) denote the minimum number of edges in an n-vertex k-critical graph. In a very recent paper, we gave a lower bound, f k (n)≥(k, n), that is sharp for every n≡1 (mod k−1). It is also sharp for k=4 and every n≥6. In this note, we present a simple proof of the bound for k=4. It implies the case k=4 of two conjectures: Gallai in 1963 conjectured that if n≡1 (mod k−1) then \(f_k (n)\tfrac{{(k + 1)(k - 2)n - k(k - 3)}} {{2(k - 1)}}\), and Ore in 1967 conjectured that for every k≥4 and \(n \geqslant k + 2,f_k (n + k - 1) = f(n) + \tfrac{{k - 1}} {2}(k - \tfrac{2} {{k - 1}})\). We also show that our result implies a simple short proof of Grötzsch’s Theorem that every triangle-free planar graph is 3-colorable.

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Correspondence to Alexandr Kostochka.

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Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

Research of this author is partially supported by the Arnold O. Beckman Research Award of the University of Illinois at Urbana-Champaign and from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.”

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Kostochka, A., Yancey, M. Ore’s conjecture for k=4 and Grötzsch’s Theorem. Combinatorica 34, 323–329 (2014). https://doi.org/10.1007/s00493-014-3020-x

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Keywords

  • Planar Graph
  • Simple Proof
  • Chromatic Number
  • Proper Coloring
  • Graph Coloring Problem