Remarks on a Conjecture of Barát and Tóth

摘要: In 2010,  Barat and Toth 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 bound $r(c)$ such $r$, where $r \geq r(c)$, on $r+c$ vertices note, we verify validity of this conjecture $c=5$, show counterexamples all $c 6$.