Extremal subgraphs with respect to vertex degree bounds

A weighted graph (G,w) is a graph G=(V,E) together with a weight function on its vertices w : V→N0. We consider the maximal (minimal) number α0,w(G) (β0,w(G)) of edges in a spanning subgraph H of G with d(v,H)⩽w(v) (d(v,H)⩾w(v)) for all v∈V. For these parameters and their corresponding duals we prove—among other results—generalizations of the well-known theorems of König (Mat. Fiz. Lapok 38 (1931) 116–119) and Gallai (Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2 (1959) 133–138). Note that these notions naturally generalize maximum matchings and minimum edge covers in graphs.