Decomposition of toroidal graphs without some subgraphs

TL;DR

This paper presents a unified method to prove that a broad class of toroidal graphs without certain cycles can be decomposed into a subgraph with bounded degree and a 2-degenerate graph, extending previous specific results.

Contribution

It introduces a unified approach to show that a superclass of toroidal graphs without certain cycles are all $(2, 1)$-decomposable, generalizing earlier findings.

Findings

A common superclass of $ extit{T}_{i,j}$ graphs is $(2,1)$-decomposable.

The approach simplifies proofs for multiple cycle-restricted toroidal graphs.

Extends known decomposability results to a broader class of graphs.

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 $\Delta(H) \leq h$ 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.