Decide if there is a sbugroup of edges in graph that every vertex meets exactly k edges from subgroup
The question is a bit misleading, since the proposed algorithm finds whether there is a "multisubset" of the edges set which is $k$-regular &mdash; that is, you are allowed to take the same edge several times, if it helps. Deciding whether the graph contains a bona fide $k$-factor (a $k$-regular subgraph which touches all vertices) is much more tricky &mdash; see Meijer, Núñez-Rodríguez, and Rappaport, An algorithm for computing simple $k$-factors.