The maximum-degree conjecture for critical graphs

A graph is critical with tree-depth $k$ if its tree-depth is $k$ and every proper minor has smaller tree-depth. The maximum degree of a graph $G$ is denoted by $\delta(G)$.

The maximum-degree conjecture. Every critical graph with tree-depth $k$ has maximum degree at most

$$k-1.$$

The bound is proved for the class of $1$-unique critical graphs, and several classes of critical graphs are known to lie in that class. Whether every critical graph satisfies the bound remains open.