Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 3, June 2022



Page(s)  185  188  
DOI  https://doi.org/10.1051/wujns/2022273185  
Published online  24 August 2022 
Mathematics
CLC number: O 157.5
Gracefulness of Two Kinds of Unconnected Graphs with Even Vertices
College of Science, North China Institute of Science and Technology, Sanhe
065201, Hebei, China
Received:
10
February
2022
Two kinds of unconnected double fan graphs with even vertices, and were presented. For natural number ,,the two graphs are all graceful graphs, where are evenvertices path, is oddvertices path, are the complement of graph , is the join graph of and .
Key words: unconnected graph / double fan graph / graceful graph / graceful label / even vertices
Biography: WEN Xiaoyan, female, Lecturer, research direction: graph theory. Email: xiaoyanwen_ncist@126.com
Foundation item: Supported by the National Natural Science Foundation of China(11702094) and the Fundamental Research Funds for the Central University(3142015045)
© Wuhan University 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^{[17]}. 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 nongraceful graphs has rarely been studied, thus it is rather significant to have further research on the gracefulness of fan graph and nongraceful 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 kgraceful graph, call f a kgraceful label of graph . 1graceful graph is also called graceful graph, and 1graceful 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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