Let G be a graph which may have multiple edges but no loops, \(\mu _{G}(v)\) be the multiplicity of G (which is the maximum among the numbers of edges between v and the other vertex on G), and a and b be any two functions, s.t. \(a,b: V(G)\rightarrow \mathbb {N} \backslash \{0,1\}\) . A theta is a graph made of three internally vertex-disjoint chordless paths \(P_{1}=x- \cdots -y\) , \(P_{2}=x-\cdots -y\) , \(P_{3}=x-\cdots -y\) of length at least two and no edges exist between \(P_i\) and \(P_j\) ( \(i\ne j,i,j\in \{1,2,3\}\) ) except the three edges incident to x and the three edges incident to y. For a partition (X, Y) of V(G), and any \(x\in X\) , \(y\in Y\) , let \(d_X(x)\) denote by the degree of x in G[X], and \(d_Y(y)\) be defined similarly. In this paper, we show that a graph G admits a partition (A, B) such that \(d_{A}(x)\ge a(x)\) for any \(x\in A\) and \(d_{B}(y) \ge b(y)\) for any \(y\in B\) if G is \(\mathcal {H}\) -free and \(d_{G}(v)\ge a + b + 2\mu _{G}(v)-3\) for any vertex \(v\in V(G)\) , where for each member \(H \in \mathcal {H}\) , the underlying of H belongs to {triangle, wheel, theta}.
