# Gromov Hyperbolicity in Mycielskian Graphs

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

## 2. Definitions and Background

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**1.**

**Proof.**

## 3. Proof of the Main Parts of Theorem 1

**Theorem**

**6.**

**Proof.**

**Lemma**

**7.**

- (i)
- If $x,y\in G$ and ${d}_{G}(x,y)\le 9/2$, then ${d}_{{G}^{M}}(x,y)={d}_{G}(x,y)$.
- (ii)
- If $x,y\in J(G)\setminus V(G)$ and ${d}_{G}(x,y)\le 5$, then ${d}_{{G}^{M}}(x,y)={d}_{G}(x,y)$.

**Proof.**

**Lemma**

**8.**

**Proof.**

- If ${v}_{i},{v}_{j}\in V(G)$, then clearly ${d}_{G}({v}_{i},{v}_{j})\le diamV(G)\le 4$, and by Lemma 7, we conclude ${d}_{{G}^{M}}({v}_{i},{v}_{j})={d}_{G}({v}_{i},{v}_{j})$.
- If ${u}_{i},{u}_{j}\in V({G}^{\prime})\setminus V(G)$, then ${d}_{{G}^{M}}({u}_{i},{u}_{j})=2\le k$.
- If ${v}_{i}\in V(G)$ and ${u}_{j}\in V({G}^{\prime})\setminus V(G)$, then ${d}_{{G}^{M}}({v}_{i},{u}_{i})=2\le k$ and ${d}_{{G}^{M}}({v}_{i},{u}_{j})\le {d}_{G}({v}_{i},{v}_{j})\le k$ (if $i\ne j$).
- If $\alpha \in V({G}^{\prime})$, then ${d}_{{G}^{M}}(\alpha ,w)\le 2\le k$.

**Theorem**

**7.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

## 4. Proof of Theorem 2

**Lemma**

**9.**

**Proof.**

**Definition**

**1.**

**Remark**

**1.**

- If $\gamma =\left[xy\right]$ is a geodesic in ${G}^{M}$ and $\gamma \subset {G}^{\prime}$, there exists a geodesic $[\Pi (x)\Pi (y)]\subseteq \Pi ([xy])$; in general, equality does not hold, since two different edges in γ might project onto the same edge in $E(G)$ (for instance, if $\gamma =\{{v}_{i},{v}_{j}\}\cup \{{v}_{j},{u}_{i}\}$, then $\Pi (\gamma )=\{{v}_{i},{v}_{j}\}$; note that $l(\gamma )=2$); therefore, $l(\Pi (\gamma ))\le l(\gamma )$.
- If $l([xy])\le 1$ and $x\in V(G)$, then $[\Pi (x)\Pi (y)]=\Pi ([xy])$.
- If ${T}^{M}$ is a geodesic triangle in ${G}^{M}$ and $\Pi ({T}^{M})$ is a geodesic triangle in G and ${G}^{M}$, then $\Pi ({T}^{M})$ does not need to be a cycle even if ${T}^{M}$ is. For instance, if ${\gamma}_{1}=\{{v}_{i},{v}_{j}\}\cup \{{v}_{j},{v}_{k}\}$ and ${\gamma}_{2}=\{{v}_{i},{u}_{j}\}\cup \{{u}_{j},{v}_{k}\}$, then ${T}^{M}=\{{\gamma}_{1},{\gamma}_{2}\}$ is a geodesic bigon in ${G}^{M}$, $\Pi ({\gamma}_{1})=\Pi ({\gamma}_{2})={\gamma}_{1}$ are geodesics in G, but $\Pi ({T}^{M})$ is not a cycle, although ${T}^{M}$ is; note that $l({T}^{M})=4$ (see Figure 2).

**Lemma**

**10.**

**Proof.**

**Proof**

**of**

**Theorem**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

## 5. Proof of Theorem 3

**Proof.**

## 6. Hyperbolicity Constant for Some Particular Mycielskian Graphs and the Proof of Theorems 1, 4 and 5

**Proposition**

**3.**

**Proof.**

**Remark**

**4.**

**Corollary**

**6.**

**Corollary**

**7.**

**Corollary**

**8.**

- (i)
- If $diamV(G)=2$, then $5/4\le \delta ({G}^{M})\le 3/2$.
- (ii)
- If $diamV(G)=3$, then $3/2\le \delta ({G}^{M})\le 2$.

**Proposition**

**4.**

**Proof.**

**Remark**

**5.**

**Proposition**

**5.**

**Proof.**

**Remark**

**6.**

**Corollary**

**9.**

**Proof**

**of**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**5.**

**Proposition**

**6.**

**Proof.**

## 7. The Case of 5/4

**Lemma**

**11.**

**Theorem**

**10.**

**Proof.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Granados, A.; Pestana, D.; Portilla, A.; Rodríguez, J.M.
Gromov Hyperbolicity in Mycielskian Graphs. *Symmetry* **2017**, *9*, 131.
https://doi.org/10.3390/sym9080131

**AMA Style**

Granados A, Pestana D, Portilla A, Rodríguez JM.
Gromov Hyperbolicity in Mycielskian Graphs. *Symmetry*. 2017; 9(8):131.
https://doi.org/10.3390/sym9080131

**Chicago/Turabian Style**

Granados, Ana, Domingo Pestana, Ana Portilla, and José M. Rodríguez.
2017. "Gromov Hyperbolicity in Mycielskian Graphs" *Symmetry* 9, no. 8: 131.
https://doi.org/10.3390/sym9080131