On almost -degenerate -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 almost d -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.