A Brooks-Type Result for Sparse Critical Graphs

Abstract

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k-1)-colorable. Let fk(n) denote the minimum number of edges in an n-vertex k-critical graph. Recently the authors gave a lower bound, \({f_k}\left( n \right) \geqslant \left\lceil {\frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left[ {k - 3} \right]}}{{2\left( {k - 1} \right)}}} \right\rceil \), that solves a conjecture by Gallai from 1963 and is sharp for every n≡1 (mod k-1). It is also sharp for k=4 and every n≤6. In this paper we refine the result by describing all n-vertex k-critical graphs G with \(\left| {E\left( G \right)} \right| \geqslant \frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left( {k - 3} \right)}}{{2\left( {k - 1} \right)}}\). In particular, this result implies exact values of f5(n) for n≤7.

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Correspondence to Alexandr Kostochka.

Additional information

Research of this author is supported in part by NSF grants DMS-1266016 and DMS- 1600592 and by grants 15-01-05867 and 16-01-00499 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.

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Kostochka, A., Yancey, M. A Brooks-Type Result for Sparse Critical Graphs. Combinatorica 38, 887–934 (2018). https://doi.org/10.1007/s00493-017-3068-3

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C35