  A quick proof of Gallai's celebrated theorem on color-critical graphs is
given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms
of partitioning the vertex-set into a minimum number of hyperedges of a
hereditary hypergraph, generalizing the chromatic number. We then show examples
of applying the results to new problems and indicate the way to algorithms and
refined complexity results for all these examples at the same time.
