# Graph Colorings and Labelings Having Multiple Restrictive Conditions in Topological Coding

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## Abstract

**:**

## 1. Introduction

#### 1.1. An Example of Topological Cryptosystems

#### 1.2. Definitions

- A symbol $[a,b]$ stands for a consecutive set $\{a,a+1,\dots ,b\}$ with integers $a,b$ holding $0\le a<b$; ${[a,b]}^{o}$ denotes an odd-set$\{a,a+2,\dots ,b\}$ with odd integers $a,b$ with respect to $1\le a<b$; ${[\alpha ,\beta ]}^{e}$ is an even-set$\{\alpha ,\alpha +2,\dots ,\beta \}$ with even integers $\alpha ,\beta $.
- The cardinality of a set X is denoted as $\left|X\right|$.
- The number ${deg}_{G}\left(v\right)=\left|N\left(v\right)\right|$ is called the degree of the vertex v, where $N\left(v\right)$ is the set of neighbors of the vertex v. If ${deg}_{G}\left(v\right)=1$. we call the vertex v a leaf.
- G is a $(p,q)$-graph having p vertices and q edges.

**Definition**

**1**

**.**Let a connected $(p,q)$-graph G with $1\le p-1\le q$ admit a mapping $\alpha :V\left(G\right)\to \{0,1,2,\dots \}$. For each $xy\in E\left(G\right)$, the labelings are defined as $\alpha \left(xy\right)=\left|\alpha \right(x)-\alpha (y\left)\right|$, and we write the vertex color set by $\alpha \left(V\right(G\left)\right)=\left\{\alpha \right(u):u\in V(G\left)\right\}$ and the edge color set as $\alpha \left(E\right(G\left)\right)=\left\{\alpha \right(xy):xy\in E(G\left)\right\}$. We have the following restrictions:

- (1)
- $\left|\alpha \right(V\left(G\right)\left)\right|=p$;
- (2)
- $\alpha \left(V\right(G\left)\right)\subseteq [0,q]$, $min\alpha \left(V\right(G\left)\right)=0$;
- (3)
- $\alpha \left(V\right(G\left)\right)\subset [0,2q-1]$, $min\alpha \left(V\right(G\left)\right)=0$;
- (4)
- $\alpha \left(E\right(G\left)\right)=\left\{\alpha \right(xy):xy\in E(G\left)\right\}=[1,q]$;
- (5)
- $\alpha \left(E\left(G\right)\right)=\{\alpha \left(xy\right):xy\in E\left(G\right)\}={[1,2q-1]}^{o}$;
- (6)
- $\alpha \left(uv\right)=\alpha \left(u\right)+\alpha \left(v\right)\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}2q\right)$;
- (7)
- G is a bipartite graph with the bipartition $(X,Y)$ such that $max\left\{\alpha \right(x):x\in X\}<min\left\{\alpha \right(y):y\in Y\}$ ($\alpha \left(X\right)<\alpha \left(Y\right)$ for short).

**Definition**

**2**

**.**A total labeling $\alpha :V\left(G\right)\cup E\left(G\right)\to [1,p+q]$ for a bipartite $(p,q)$-graph G is a bijection holding:

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

**Definition**

**5.**

## 2. Connections between 6C-Labeling and Other Labelings

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A set-ordered odd-graceful topological public key H and its own topological private keys ${H}_{1},{H}_{2},{H}_{3}$.

**Figure 2.**Examples The graphs $T,{G}_{1},{G}_{2},{G}_{3}$ for illustrating the 6C-labeling defined in Definition 2, the graphs $T{H}_{1},{H}_{2},{H}_{3}$ reciprocal-inverse labeling defined in Definition 3, and the 6C-complimentary matching defined in Definition 4.

**Figure 3.**Examples for illustrating Corollary 2. (

**a**–

**d**) are T, $T{\phantom{\rule{0.166667em}{0ex}}}^{\prime}$, $T{\phantom{\rule{0.166667em}{0ex}}}^{\u2033}$, and $T{\phantom{\rule{0.166667em}{0ex}}}^{\prime}\diamond T{\phantom{\rule{0.166667em}{0ex}}}^{\u2033}$, respectively.

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**MDPI and ACS Style**

Zhang, X.; Ye, C.; Zhang, S.; Yao, B.
Graph Colorings and Labelings Having Multiple Restrictive Conditions in Topological Coding. *Mathematics* **2022**, *10*, 1592.
https://doi.org/10.3390/math10091592

**AMA Style**

Zhang X, Ye C, Zhang S, Yao B.
Graph Colorings and Labelings Having Multiple Restrictive Conditions in Topological Coding. *Mathematics*. 2022; 10(9):1592.
https://doi.org/10.3390/math10091592

**Chicago/Turabian Style**

Zhang, Xiaohui, Chengfu Ye, Shumin Zhang, and Bing Yao.
2022. "Graph Colorings and Labelings Having Multiple Restrictive Conditions in Topological Coding" *Mathematics* 10, no. 9: 1592.
https://doi.org/10.3390/math10091592