# Hamiltonicity of Token Graphs of Some Join Graphs

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Hamiltonicity in Token Graphs

#### 1.2. Basic Definitions and Results

**Theorem**

**1.**

**Theorem**

**2.**

**Corollary**

**1.**

## 2. Proof of Theorem 1

- Case $m=1.$For $n=2$ we have ${F}_{1,2}^{\left\{2\right\}}\simeq {F}_{1,2}$, and so ${F}_{1,2}^{\left\{2\right\}}$ is Hamiltonian. Now we work the case $n\ge 3$. For $1\le i<n$ let$${T}_{i}:=\{{v}_{i},{w}_{1}\}\{{v}_{i},{v}_{i+1}\}\{{v}_{i},{v}_{i+2}\}\cdots \{{v}_{i},{v}_{n}\}$$Let$$C:=\left\{\begin{array}{cc}\overleftarrow{{T}_{1}}\phantom{\rule{0.166667em}{0ex}}{T}_{2}\overleftarrow{{T}_{3}}\phantom{\rule{0.166667em}{0ex}}{T}_{4}\cdots \overleftarrow{{T}_{n-1}}\phantom{\rule{0.166667em}{0ex}}{T}_{n}\{{v}_{1},{v}_{n}\}\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{even},\hfill \\ \overleftarrow{{T}_{1}}\phantom{\rule{0.166667em}{0ex}}{T}_{2}\overleftarrow{{T}_{3}}\phantom{\rule{0.166667em}{0ex}}{T}_{4}\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}{T}_{n-1}\phantom{\rule{0.166667em}{0ex}}\overleftarrow{{T}_{n}}\phantom{\rule{0.166667em}{0ex}}\{{v}_{1},{v}_{n}\}\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}.\hfill \end{array}\right.$$We are going to show that C is a Hamiltonian cycle of ${F}_{1,n}^{\left\{2\right\}}$. Suppose that n is even, so$$C=\underset{\overleftarrow{{T}_{1}}}{\underbrace{\{{v}_{1},{v}_{n}\}\cdots \{{v}_{1},{w}_{1}\}}}\underset{{T}_{2}}{\underbrace{\{{v}_{2},{w}_{1}\}\cdots \{{v}_{2},{v}_{n}\}}}\phantom{\rule{0.166667em}{0ex}}\cdots \phantom{\rule{0.166667em}{0ex}}\underset{{T}_{n}}{\underbrace{\{{v}_{n},{w}_{1}\}}}\{{v}_{1},{v}_{n}\}.$$For i odd, the final vertex of $\overleftarrow{{T}_{i}}$ is $\{{v}_{i},{w}_{1}\}$, while the initial vertex of ${T}_{i+1}$ is $\{{v}_{i+1},{w}_{1}\}$, and since these two vertices are adjacent in ${F}_{1,n}^{\left\{2\right\}}$, the concatenation $\overleftarrow{{T}_{i}}\phantom{\rule{0.166667em}{0ex}}{T}_{i+1}$ corresponds to a path in ${F}_{1,n}^{\left\{2\right\}}$. Similarly, for i even, the final vertex of ${T}_{i}$ is $\{{v}_{i},{v}_{n}\}$ while the initial vertex of $\overleftarrow{{T}_{i+1}}$ is $\{{v}_{i+1},{v}_{n}\}$, so again, the concatenation ${T}_{i}\phantom{\rule{0.166667em}{0ex}}\overleftarrow{{T}_{i+1}}$ corresponds to a path in ${F}_{1,n}^{\left\{2\right\}}$. We also note that the unique vertex of ${T}_{n}$ is $\{{v}_{n},{w}_{1}\}$, which is adjacent to $\{{v}_{1},{v}_{n}\}$. As the initial vertex of $\overleftarrow{{T}_{1}}$ is $\{{v}_{1},{v}_{n}\}$, we have that C is a cycle in ${F}_{1,n}^{\left\{2\right\}}$. As an example, in Figure 3 we show the Hamiltonian cycle C in ${F}_{1,5}^{\left\{2\right\}}$, which is constructed as above.The proof for n odd is analogous.
- Case $m=2n.$Let C be the cycle defined in the previous case depending on the parity of n. Let$${P}_{1}:=\{{v}_{n},{w}_{1}\}\stackrel{C}{\to}\{{v}_{1},{v}_{n}\}$$$$\begin{array}{cc}\hfill {P}_{i}:=& \{{w}_{i},{v}_{n}\}\{{w}_{i},{w}_{1}\}\{{w}_{i},{v}_{n-1}\}\{{w}_{i},{w}_{i+(n-1)}\}\{{w}_{i},{v}_{n-2}\}\{{w}_{i},{w}_{i+(n-2)}\}\{{w}_{i},{v}_{n-3}\}\hfill \\ & \{{w}_{i},{w}_{i+(n-3)}\}\dots \{{w}_{i},{v}_{2}\}\{{w}_{i},{w}_{i+2}\}\{{w}_{i},{v}_{1}\}\{{w}_{i},{w}_{i+1}\}.\hfill \end{array}$$We can observe that after $\{{w}_{i},{w}_{1}\}$, the vertices in the path ${P}_{i}$ follows the pattern $\{{w}_{i},{v}_{j}\}\{{w}_{i},{w}_{i+j}\}$, from $j=n-1$ to 1. For $n+1\le i\le 2n$ let$$\begin{array}{cc}\hfill {P}_{i}:=& \{{w}_{i},{v}_{n}\}\{{w}_{i},{w}_{i+n}\}\{{w}_{i},{v}_{n-1}\}\{{w}_{i},{w}_{i+(n-1)}\}\{{w}_{i},{v}_{n-2}\}\{{w}_{i},{w}_{i+(n-2)}\}\dots \hfill \\ & \{{w}_{i},{v}_{2}\}\{{w}_{i},{w}_{i+2}\}\{{w}_{i},{v}_{1}\}\{{w}_{i},{w}_{i+1}\},\hfill \end{array}$$Let$${C}_{2}:={P}_{1}\phantom{\rule{0.166667em}{0ex}}{P}_{2}\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}{P}_{2n}\{{v}_{n},{w}_{1}\}.$$Let us show that ${C}_{2}$ is a Hamiltonian cycle of ${F}_{m,n}^{\left\{2\right\}}$. First we show that$$\{V\left({P}_{1}\right),\cdots ,V\left({P}_{2n}\right)\}$$
- -
- $\{{v}_{i},{v}_{j}\}$ belongs to ${P}_{1}$, for any $i,j\in \left[n\right]$ with $i\ne j$.
- -
- $\{{w}_{i},{v}_{j}\}$ belongs to ${P}_{i}$, for any $i\in \left[m\right]$ and $j\in \left[n\right]$.
- -
- $\{{w}_{i},{w}_{1}\}$ belongs to ${P}_{i}$, for any $i\in \left[m\right]$ with $i\ne 1$.
- -
- Consider now the vertices of type $\{{w}_{i},{w}_{j}\}$, for $1<i<j\le n$,
- *
- $\{{w}_{i},{w}_{j}\}$ belongs to ${P}_{i}$, for any $1<i\le n$ and $i<j\le i+n-1$.
- *
- $\{{w}_{i},{w}_{j}\}$ belongs to ${P}_{j}$, for any $1<i\le n$ and $i+n-1<j\le 2n$.
- *
- $\{{w}_{i},{w}_{j}\}$ belongs to ${P}_{i}$, for any $n<i<2n$ and $i<j\le 2n$.

Thus, $\{V\left({P}_{1}\right),\cdots ,V\left({P}_{2n}\right)\}$ is a partition of $V\left({F}_{m,n}^{\left\{2\right\}}\right)$. Next we show that ${C}_{2}$ is a cycle. We observe that- (1)
- ${P}_{i}$ induces a path in ${F}_{m,n}^{\left\{2\right\}}$, for each $i\in \left[2n\right]$;
- (2)
- the final vertex of ${P}_{1}$ is $\{{v}_{1},{v}_{n}\}$, while the initial vertex of ${P}_{2}$ is $\{{w}_{2},{v}_{n}\}$, and these two vertices are adjacent in ${F}_{m,n}^{\left\{2\right\}}$;
- (3)
- for i with $1<i<2n$, the final vertex of ${P}_{i}$ is $\{{w}_{i},{w}_{i+1}\}$ while the initial vertex of ${P}_{i+1}$ is $\{{w}_{i+1},{v}_{n}\}$, and these two vertices are adjacent in ${F}_{m,n}^{\left\{2\right\}}$; and
- (4)
- the final vertex of ${P}_{2n}$ is $\{{w}_{1},{w}_{n}\}$ while the initial vertex of ${P}_{1}$ is $\{{v}_{n},{w}_{1}\}$, and these two vertices are adjacent in ${F}_{m,n}^{\left\{2\right\}}$.

Statements (1)–(4) together imply that ${C}_{2}$ is a cycle in ${F}_{m,n}^{\left\{2\right\}}$. Thus, ${C}_{2}$ is a Hamiltonian cycle of ${F}_{m,n}^{\left\{2\right\}}$. Note that the vertices $\{{w}_{1},{v}_{1}\}$ and $\{{v}_{1},{v}_{2}\}$ are adjacent in ${C}_{2}$, since they are adjacent in ${P}_{1}$.As an example, in Figure 4 we show the Hamiltonian cycle ${C}_{2}$ in the graph ${F}_{4,2}^{\left\{2\right\}}$.

- Case $1<m<2\phantom{\rule{0.166667em}{0ex}}n.$Consider again the paths ${P}_{1},\cdots ,{P}_{m}$ defined in the previous case and let us modify them slightly in the following way:
- -
- ${P}_{1}^{\prime}={P}_{1}$;
- -
- for $1<i<m$, let ${P}_{i}^{\prime}$ be the path obtained from ${P}_{i}$ by deleting the vertices of type $\{{w}_{i},{w}_{j}\}$, for each $j>m$;
- -
- let ${P}_{m}^{\prime}$ be the path obtained from ${P}_{m}$ by first interchanging the vertices $\{{w}_{m},{w}_{m+1}\}$ and $\{{w}_{m},{w}_{1}\}$ from their current positions in ${P}_{m}$, and then deleting the vertices of type $\{{w}_{m},{w}_{j}\}$, for every $j>m$.

Given this construction of ${P}_{i}^{\prime}$ we have the following:- (A1)
- ${P}_{i}^{\prime}$ induces a path in ${F}_{m,n}^{\left\{2\right\}}$;
- (A2)
- for $1\le i<m$ the path ${P}_{i}^{\prime}$ has the same initial and final vertices as the path ${P}_{i}$, and ${P}_{m}^{\prime}$ has the same initial vertex as ${P}_{m}$, and its final vertex is $\{{w}_{i},{w}_{1}\}$;
- (A3)
- since we have deleted only the vertices of type $\{{w}_{i},{w}_{j}\}$ from ${P}_{i}$ to obtain ${P}_{i}^{\prime}$, for each $j>m$ and $i\in \left[m\right]$, it follows that $\{V\left({P}_{1}^{\prime}\right),\dots ,V\left({P}_{m}^{\prime}\right)\}$ is a partition of $V\left({F}_{m,n}^{\left\{2\right\}}\right)$.

By (A1) and (A2) we can concatenate the paths ${P}_{1}^{\prime},\dots ,{P}_{m}^{\prime}$ into a cycle ${C}^{\prime}$ as follows:$${C}^{\prime}:={P}_{1}^{\prime}\phantom{\rule{0.166667em}{0ex}}{P}_{2}^{\prime}\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}{P}_{m}^{\prime}({v}_{n},{w}_{1})$$ - Case $m>2\phantom{\rule{0.166667em}{0ex}}n.$Here, our aim is to show that ${F}_{m,n}^{\left\{2\right\}}$ is not Hamiltonian by using the following known result posed in West’s book [29].

**Proposition**

**1**

## 3. Proof of Theorem 2

**Proposition**

**2.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**$n-k$ is odd:**

**Remark**

**3.**

- (1)
- the final vertex of $\overleftarrow{{P}_{k}}$ is the vertex ${Z}_{k-1}=\{{v}_{0},{v}_{1},\cdots ,{v}_{k-2},{v}_{k}\}$, while the initial vertex of ${P}_{k+1}$ is the vertex ${X}_{k+1}=\{{v}_{0},{v}_{1},\cdots ,{v}_{k-2},{v}_{k+1}\}$, and these two vertices are adjacent in ${F}_{1,n}^{\left\{k\right\}}$;
- (2)
- for i with $k+1\le i\le n-1$, the final vertex of ${P}_{i}$ is the vertex ${Y}_{i}=$$\{{v}_{1},{v}_{2},\cdots ,{v}_{k-1},{v}_{i}\}$, while the initial vertex of $\overleftarrow{{P}_{i+1}}$ is the vertex ${Y}_{i+1}=$$\{{v}_{1},{v}_{2},\cdots ,{v}_{k-1},{v}_{i+1}\}$, and these two vertices are adjacent in ${F}_{1,n}^{\left\{k\right\}}$;
- (3)
- for i with $k+1\le i\le n-1$, the final vertex of $\overleftarrow{{P}_{i}}$ is ${X}_{i}=\{{v}_{0},{v}_{1},\cdots ,{v}_{k-2},{v}_{i}\}$, while the initial vertex of ${P}_{i+1}$ is ${X}_{i+1}=\{{v}_{0},{v}_{1},\cdots ,{v}_{k-2},{v}_{i+1}\}$, and these two vertices are adjacent in ${F}_{1,n}^{\left\{k\right\}}$;
- (4)
- finally, the final vertex of ${P}_{n}$ is the vertex ${Y}_{n}=\{{v}_{1},{v}_{2},\cdots ,{v}_{k-1},{v}_{n}\}$ while the initial vertex of $\overleftarrow{{P}_{k}}$ is the vertex ${Z}_{k}=\{{v}_{0},{v}_{1},\cdots ,{v}_{k-1}\}$, and these two vertices are adjacent in ${F}_{1,n}^{\left\{k\right\}}$.

**$n-k$ is even:**

**Remark**

**4.**

**Proof**

**of Theorem 2.**

- (i)
- ${C}_{1}$ is a Hamiltonian cycle of ${H}_{1}$, where the vertices ${X}_{1}:=\{{w}_{1},{w}_{2},{v}_{1},{v}_{2},\dots ,{v}_{k-2}\}$ and ${Y}_{1}:=\{{w}_{1},{v}_{1},{v}_{2},\dots ,{v}_{k-1}\}$ are adjacent in ${C}_{1}$;
- (ii)
- ${C}_{2}$ is a Hamiltonian cycle of ${H}_{2}$, where the vertices ${X}_{2}:=\{{w}_{2},{v}_{1},{v}_{2},\dots ,{v}_{k-1}\}$ and ${Y}_{2}:=\{{v}_{1},{v}_{2},\dots ,{v}_{k}\}$ are adjacent in ${C}_{2}$.

## 4. A Relationship between Gray Codes for Combinations and the Hamiltonicity of Token Graphs

- (1)
- The transposition condition: two k-subsets are close if they differ in exactly two elements. Example: $\{1,2,5\}$ and $\{2,4,5\}$ are close, while $\{1,2,5\}$ and $\{1,3,4\}$ are not.
- (2)
- The adjacent transposition condition: two k-subsets are close if they differ in exactly two consecutive elements i and $i+1$. Example: $\{1,2,5\}$ and $\{1,3,5\}$ are close, while $\{1,2,5\}$ and $\{1,4,5\}$ are not.
- (3)
- The one or two apart transposition condition: two k-subsets are close if they differ in exactly two elements i and j, with $|i-j|\le 2$. Example: $\{1,2,5\}$ and $\{1,4,5\}$ are close, while $\{1,2,5\}$ and $\{2,4,5\}$ are not.

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 5. Conclusions

- 1.
- To find other families of graphs with Hamiltonian k-token graphs.

- 2.
- Given two graphs G and H, consider the Cartesian product $G\phantom{\rule{0.166667em}{0ex}}\square \phantom{\rule{0.166667em}{0ex}}H$ of G and H. To study the Hamiltonicity of ${(G\phantom{\rule{0.166667em}{0ex}}\square \phantom{\rule{0.166667em}{0ex}}H)}^{\left\{k\right\}}$ in terms of the Hamiltonicity of G and H. Similarly for other products of graphs as the corona of two graphs.

- 3.
- For $k>2$, to find the smallest Hamiltonian graph G for which ${G}^{\left\{k\right\}}$ is Hamiltonian.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Adame, L.E.; Rivera, L.M.; Trujillo-Negrete, A.L. Hamiltonicity of Token Graphs of Some Join Graphs. *Symmetry* **2021**, *13*, 1076.
https://doi.org/10.3390/sym13061076

**AMA Style**

Adame LE, Rivera LM, Trujillo-Negrete AL. Hamiltonicity of Token Graphs of Some Join Graphs. *Symmetry*. 2021; 13(6):1076.
https://doi.org/10.3390/sym13061076

**Chicago/Turabian Style**

Adame, Luis Enrique, Luis Manuel Rivera, and Ana Laura Trujillo-Negrete. 2021. "Hamiltonicity of Token Graphs of Some Join Graphs" *Symmetry* 13, no. 6: 1076.
https://doi.org/10.3390/sym13061076