Decomposing multigraphs into two Class I subgraphs with prescribed maximum degrees

Abstract

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