Remarks on a Conjecture of Barát and Tóth

  • Atílio G. Luiz
  • R. Bruce Richter
Keywords: Colour-critical graphs, Hajós conjecture, Albertson conjecture


In 2010,  Barát and Tóth verified that any $r$-critical graph with at most $r+4$ vertices has a subdivision of $K_r$. Based in this result, the authors conjectured that, for every positive integer $c$, there exists a bound $r(c)$ such that for any $r$, where $r \geq r(c)$, any $r$-critical graph on $r+c$ vertices has a subdivision of $K_r$. In this note, we verify the validity of this conjecture for $c=5$, and show counterexamples for all $c \geq 6$.
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