Decomposing multigraphs into two Class I subgraphs with prescribed maximum degrees

Let $G$ be a finite loopless multigraph with maximum degree $\mathrm{\Delta }$ and ${\chi }^{\mathrm{\prime }}(G)=\mathrm{\Delta }+k$, where $k\ge 0$. We prove that for every pair of positive integers $p,q$ satisfying $$p+q=\mathrm{\Delta }+k\text{and}p,q\le \mathrm{\Delta },$$ the graph $G$ admits an edge decomposition into two spanning Class I subgraphs $H_p,H_q$ with $$\mathrm{\Delta }(H_p)=p\text{and}\mathrm{\Delta }(H_q)=q\mathrm{.}$$