A Brooks-Type Result for Sparse Critical Graphs



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. 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 \(\left| {E\left( G \right)} \right| = \frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left( {k - 3} \right)}}{{2\left( {k - 1} \right)}}\). In particular, this result implies exact values of f5(n) for n≥7.

Mathematics Subject Classification (2000)

05C15 05C35 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Department of MathematicsUniversity of IllinoisUrbanaUSA

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