Among some results, we prove the following two theorems, which are generalizations of known results. Let G be a connected graph and k be a positive even integer. (i) If G is k-tough, then for any set W of even number of vertices of G, G has a factor F such that \(\deg _F(x) \in \{1,3, 5, \ldots , k+1\}\) for all \(x\in W\) and \(\deg _F(y)=k\) for all \(y\in V(G)-W\) . (ii) For any set W of even number of vertices of G, G has a factor F such that \(\deg _F(x) \equiv 1 \pmod {2}\) for all \(x\in W\) and \(\deg _F(y)\in \{k, k+2, k+4, \ldots \}\) for all \(y\in V(G)-W\) if and only if \(\deg _G(X) - k|X| - \omega (G-X) \ge -1\) for all \(X \subset V(G)\) , where \(\omega (G-X)\) denotes the number of components of \(G-X\) .
