Barrus–Sinković conjecture on the maximum degree of critical graphs

Let $G$ be a $k$-critical graph, meaning that $G$ has tree-depth $k$ and every proper minor of $G$ has tree-depth less than $k$.

Barrus–Sinković conjecture. $G$ has maximum degree at most

$$k-1.$$

The conjecture arose from patterns in critical graphs of small tree-depth and is attributed in the source to Barrus and Sinković. It is false infinitely often, as shown in the paper.