# The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The $(3,1)$-Decomposability of Toroidal Graphs

**Theorem**

**1.**

**Definition**

**1.**

**Lemma**

**1.**

- (1)
- there is no ${3}^{-}$-vertex in G;
- (2)
- there is no adjacent 4-vertices in G;
- (3)
- there is no minor 3-face in G.

**Proof.**

- R1
- For a 4-vertex, it transfers $\frac{1}{2}$ charge to each incident 3- and 4-face.
- R2
- For a 5-vertex, it transfers $\frac{7}{4}$ charge to each incident 3-face and $\frac{1}{2}$ charge to each incident 4-face.
- R3
- For a ${6}^{+}$-vertex, it transfers $\frac{7}{4}$ charge to each incident 3-face and $\frac{11}{12}$ charge to each incident 4-face.

## 3. The $(2,1)$-Decomposability of Toroidal Graphs

**Theorem**

**2.**

**Lemma**

**2.**

- (1)
- ${2}^{-}$-vertex;
- (2)
- a 3-vertex adjacent to another 3-vertex.

**Case**

**1.**

- R1.1
- A ${5}^{+}$-face transfers $\frac{1}{3}$ to every incident 3-vertex.

**Case**

**2.**

**Definition**

**2.**

**Lemma**

**3.**

- (1)
- a $(3,4,3,4)$-face;
- (2)
- a light 3-vertex.

**Proof.**

- R2.1
- For a 3-vertex, if it is on a 4-face, then it acquires $\frac{1}{2}$ charge from each incident ${5}^{+}$-face; otherwise, it acquires $\frac{1}{3}$ charge from each incident face.

**Case**

**3.**

- R3.1
- For a 3-face f, it acquires $\frac{1}{3}$ charge from each adjacent ${7}^{+}$-face.
- R3.2
- For a 3-vertex v, if v is on a 3-face, it acquires $\frac{1}{2}$ charge from each incident ${7}^{+}$-face; otherwise, v acquires $\frac{1}{3}$ charge from each incident face.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A minor 3-face. (

**b**) Local orientation of D, where the thick lines are edges in ${F}_{2}$.

**Figure 2.**(

**a**) A light 3-vetex. (

**b**) and (

**c**) Local orientation of D, where thick lines are edges in subgraph F.

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**MDPI and ACS Style**

Lu, H.; Li, F.
The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles. *Axioms* **2023**, *12*, 173.
https://doi.org/10.3390/axioms12020173

**AMA Style**

Lu H, Li F.
The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles. *Axioms*. 2023; 12(2):173.
https://doi.org/10.3390/axioms12020173

**Chicago/Turabian Style**

Lu, Huajing, and Fengwei Li.
2023. "The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles" *Axioms* 12, no. 2: 173.
https://doi.org/10.3390/axioms12020173