Decomposition of Toroidal Graphs Without Some Subgraphs

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