Decomposition of toroidal graphs without some subgraphs

We consider a family of toroidal graphs, denoted by ${\mathcal{T}}_{i,j}$, which contain neither $i$-cycles nor $j$-cycles. A graph $G$ is $(d,h)$-decomposable if it contains a subgraph $H$ with $\mathrm{\Delta }(H)\le h$ such that $G-E(H)$ is a $d$-degenerate graph. For each pair $(i,j)\in \{(3,4),(3,6),(4,6),(4,7)\}$, Lu and Li proved that every graph in ${\mathcal{T}}_{i,j}$ is $(2,1)$-decomposable. In this short note, we present a unified approach to prove that a common superclass of ${\mathcal{T}}_{i,j}$ is also $(2,1)$-decomposable.