Decomposition of toroidal graphs without some subgraphs

TL;DR

This note addresses toroidal graphs without a small set of subgraphs by proving they are $=(2,1)$-decomposable, i.e., admit a subgraph $H$ with $\Delta(H)\le 1$ such that the remainder is $2$-degenerate. The authors employ a unified, discharging-based proof on a minimal counterexample embedded on the torus, starting from the charge $\omega(z)=\deg(z)-4$ and ruling out forbidden configurations to obtain a contradiction. This extends prior results for the family $\mathcal{T}_{i,j}$ with $(i,j)\in\{(3,4),(3,6),(4,6),(4,7)\}$ and provides a common framework for $(d,h)$-decomposability on toroidal graphs, with implications for related defective colorings and acyclic orientations.

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.

Figures (6)

• Figure 1: Common forbidden configurations in \ref{['LLWZ']}(1) and (2).
• Figure 2: Certain forbidden configurations in \ref{['LLWZ']}(1).
• Figure 3: Certain forbidden configurations in \ref{['LLWZ']}(2).
• Figure 4: Certain forbidden configurations in \ref{['J']}.
• Figure 5: Reducible configurations.

Theorems & Definitions (8)

• Theorem 1.1: Cho et al. MR4472765
• Theorem 1.2: Lu and Zhu MR4051856
• Theorem 1.3: Li et al. MR4557782
• Theorem 1.4: Lu and Li Lu2023
• Theorem 1.5: Tian et al. MR4593925
• Theorem 1.6: Lu and Li Lu2023
• Theorem 1.7
• Lemma 2.1