The complexity of regular subgraph recognition

Abstract Given a graph G we wish to determine whether it contains a partial k -regular subgraph. In this paper it is shown that this problem is NP-complete for graphs with maximum degree k +1 when k ≥3. (The k =1,2 cases are trivially solvable in polynomial time.) The k -regular subgraph problem remains NP-complete when G is restricted to Hamiltonian or Eulerian graphs. We also examine the cases where the k -regular subgraph is further restricted to being spanning and/or connected.