Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 27, Number 3, June 2022 185 - 188 https://doi.org/10.1051/wujns/2022273185 24 August 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 0 Introduction

Research on labeling of graceful graph is on the rise following the process of computing science[1-7]. Researches on the gracefulness of fan graph, especially double fan graph, are very important for the study of graph[8]. The gracefulness of union graphs of double fan graph, linear graph and star graph has been proved[9], and the gracefulness of union graphs of double fan graph and all graceful graphs has also been proved[10]. The gracefulness of union graphs of fan graph and non-graceful graphs has rarely been studied, thus it is rather significant to have further research on the gracefulness of fan graph and non-graceful graphs.

The graphs in the following are all simple undirected graphs. Let be a graph, be the set of vertices of graph G, be the set of edges of graph G, be the order of the set A, be the path with n vertices, be the trivial graph with one vertex, be the joint graph of and , be the complementary graph of G.

Definition 1   Let be a graph, and let k be a positive integer. If there is a injection :{ }, such that for every edge induces a bijection { }, then we call a k-graceful graph, call f a k-graceful label of graph . 1-graceful graph is also called graceful graph, and 1-graceful label is called graceful label.

## 1 The Gracefulness of Graph

Theorem 1   For natural number , , is a graceful graph.

Proof   For {},,, ,,, , then , .

Define the vertex label f of graph as follows:

We shall prove that label f is a graceful label of graph .

(ⅰ) By the definition of f :

Hence mapping : {}is an injection.

(ⅱ) For every edge , let .

Then

Hence

Hence

: {} is a bijection, graph is a graceful graph.

## 2 The Gracefulness of Graph

Theorem 2   For natural number , , is a graceful graph.

Proof   For {},,

, then .

Define the vertex label f of graph as follows:

We shall prove that label f is a graceful label of graph .

(ⅰ) By the definition of f :

Hence mapping :{}}is an injection.

(ⅱ) Let

Hence

Hence

is a bijection, and graph is a graceful graph.

## 3 Conclusion

The gracefulness on two kinds of unconnected double fan graphs with even vertices, and , were proved. The vertex label f of graph was defined by the definition of graceful graph, and f was proved as injection/bijection. The graceful theorem of the union of unconnected triple graphs was proved. The research will be beneficial to the study of gracefulness on union graph of multiple graphs.

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