On almost (k−1)-degenerate (k+1)-chromatic graphs and hypergraphs

Recall that a (hyper)graph is d-degenerate if each of its nonempty subgraphs has a vertex of degree at most d. Every d-degenerate (hyper)graph is (easily) (d+1)-colorable. A (hyper)graph is almostd-degenerate if it is not d-degenerate, but each of its proper subgraphs is d-degenerate. In particular, if G is almost (k−1)-degenerate, then after deleting any edge it is k-colorable. For k≥2, we study properties of almost (k−1)-degenerate (hyper)graphs that are not k-colorable. By definition, each such (hyper)graph is (k+1)-critical.