Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?
Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?
This was conjectured by Dirac in 1970 for $k\geq 4$ and $r=1$. Dirac's conjecture was proved, for $k=5$, by Brown [Br92]. Lattanzio [La02] proved there exist such graphs for all $k$ such that $k-1$ is not prime. Independently, Jensen [Je02] gave an alternative construction for all $k\geq 5$. The case $k=4$ and $r=1$ remains open.
Martinsson and Steiner [MaSt25] proved this is true for every $r\geq 1$ if $k$ is sufficiently large, depending on $r$.
This is Problem 91 in the graph problems collection.