Résumé en anglais

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321–324] proved that if every vertex v in a graph G has degree d(v)greater-or-equal, slanteda(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that dA(v)greater-or-equal, slanteda(v) for every vset membership, variantA and dB(v)greater-or-equal, slantedb(v) for every vset membership, variantB. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7–9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237–239] strengthened this result, proving that it suffices to assume d(v)greater-or-equal, slanteda+b (a,bgreater-or-equal, slanted1) or just d(v)greater-or-equal, slanteda+b-1 (a,bgreater-or-equal, slanted2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.