# Hyperbolicity of Direct Products of Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Background

**Definition**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Definition**

**2.**

**Corollary**

**1.**

**Theorem**

**1.**

**Corollary**

**2.**

**Theorem**

**2**

## 3. Hyperbolic Direct Products

**Proposition**

**3.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Remark**

**1.**

**Lemma**

**5.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Corollary**

**4.**

**Theorem**

**10.**

**Theorem**

**11.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**12.**

**Lemma**

**7.**

**Proof.**

**Definition**

**3.**

**Remark**

**2.**

**Lemma**

**8.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**13.**

- (1)
- $DX$ is hyperbolic.
- (2)
- X is hyperbolic and there exists a constant ${c}_{1}$ such that for every $k,l\in J$ and $a\in {\eta}_{k},b\in {\eta}_{l}$ we have ${d}_{X}(x,{\cup}_{j\in J}{\eta}_{j})\le {c}_{1}$ for every $x\in \left[ab\right]\subset X$.
- (3)
- X is hyperbolic and there exist constants ${c}_{2},\alpha ,\beta $ such that for every $k,l\in J$ and $a\in {\eta}_{k},b\in {\eta}_{l}$ we have ${d}_{X}(x,{\cup}_{j\in J}{\eta}_{j})\le {c}_{2}$ for every x in some $(\alpha ,\beta )$-quasi-geodesic joining a with b in X.

**Theorem**

**14.**

**Theorem**

**15.**

- (1)
- ${G}_{1}\times {G}_{2}$ is hyperbolic.
- (2)
- ${G}_{1}$ is hyperbolic and there exists a constant ${c}_{1}$, such that for every $k,l\in J$ and ${w}_{k}\in {B}_{k}$, ${w}_{l}\in {B}_{l}$ there exists a geodesic $\left[{w}_{k}{w}_{l}\right]$ in ${G}_{1}$ with ${d}_{{G}_{1}}(x,{\cup}_{j\in J}{w}_{j})\le {c}_{1}$ for every $x\in \left[{w}_{k}{w}_{l}\right]$.
- (3)
- ${G}_{1}$ is hyperbolic and there exist constants ${c}_{2},\alpha ,\beta $, such that for every $k,l\in J$ we have ${d}_{{G}_{1}}(x,{\cup}_{j\in J}{w}_{j})\le {c}_{2}$ for every x in some ($\alpha ,\beta $)-quasi-geodesic joining ${w}_{k}$ with ${w}_{l}$ in ${G}_{1}$.

**Proof.**

**Corollary**

**5.**

## 4. Bounds for the Hyperbolicity Constant of Some Direct Products

**Theorem**

**16.**

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**9.**

**Remark**

**5.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Theorem**

**18.**

**Proof.**

**Theorem**

**19.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**If ${G}_{1}$ and ${G}_{2}$ are unbounded, for any odd n, there is a geodesic triangle $T\subset {G}_{1}\times {G}_{2}$ with $\delta \left(T\right)\ge \frac{n-3}{2}$.

**Figure 2.**Two geodesic triangles, ${T}_{1},{T}_{2}$, which are odd cycles and a geodesic g joining them define an even closed path.

**Figure 3.**If ${d}_{{G}_{1}}(m,{T}_{1}\cup g)\le 8\delta $, then $m\in \left[{x}^{\prime}{z}^{\prime}\right]$ and there is a point ${m}^{\prime}\in \left[{x}^{\prime}m\right]\subset \left[{x}^{\prime}{w}_{k}\right]$ such that ${d}_{{G}_{1}}(m,{m}^{\prime})=2(D+8\delta )$.

**Figure 4.**For any geodesic $\gamma $ in ${P}_{m}\times {P}_{n}$ with ${\pi}_{1}\left(u\right)={\pi}_{1}\left(v\right)$ for some different vertices $u,v$ in $\gamma $, then $L\left(\gamma \right)\le n-1$.

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**MDPI and ACS Style**

Carballosa, W.; De la Cruz, A.; Martínez-Pérez, A.; Rodríguez, J.M. Hyperbolicity of Direct Products of Graphs. *Symmetry* **2018**, *10*, 279.
https://doi.org/10.3390/sym10070279

**AMA Style**

Carballosa W, De la Cruz A, Martínez-Pérez A, Rodríguez JM. Hyperbolicity of Direct Products of Graphs. *Symmetry*. 2018; 10(7):279.
https://doi.org/10.3390/sym10070279

**Chicago/Turabian Style**

Carballosa, Walter, Amauris De la Cruz, Alvaro Martínez-Pérez, and José M. Rodríguez. 2018. "Hyperbolicity of Direct Products of Graphs" *Symmetry* 10, no. 7: 279.
https://doi.org/10.3390/sym10070279