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arXiv · 2502.18945

Decomposition of toroidal graphs without some subgraphs

Tao Wang·Xiaojing Yang
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Abstract

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 $Δ(H) \leq h$ 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.

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Tao Wang, Xiaojing Yang. 2025-02-26. Decomposition of toroidal graphs without some subgraphs. https://doi.org/10.1007/s40840-023-01631-x

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