We say that a graph G is (?0,?)-chromatic if Chr(G) = ?, while Chr(G?) ≤ ?0 for any subgraph G? of G of size < |G|. The main result of this paper reads as follows. If ??+CH? holds for a given uncountable cardinal ?, then for every cardinal ?≤?, there exists an (?0,?)-chromatic graph of size ?+. We also study (?0,?+)-chromatic graphs of size ?+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal ?, there exists an (?0,?+)-chromatic graph of size ?+.