Abstract
In this paper, we deduce the equivalence relationship among strongly c-elegant labelling, super-edge magic total labelling, edge antimagic total labelling, and super -magical labelling. We study some properties of the graph with a strongly c-elegant labelling. On the basis of small-scale graphs with strongly c-elegant labelling, several types of large-scale graphs are constructed through graph operations, and we further prove the existence of their strongly c-elegant labelling. In addition, we also define a transformation of strongly c-elegant labelling, which provides a method for the transformation between several strongly c-elegant labellings of a graph.
1. Introduction
Topological authentication is a new technology based on topological coding to solve the problem that text passwords are easily deciphered, and it is used for user identity authentication of smartphones [1, 2], mobile ad hoc networks [3], picture signature passwords [4], and so on. Topological authentication is an interdisciplinary research topic of graph theory, algebra, discrete mathematics, and other disciplines. Wang et al. [5, 6] designed a class of topology code composed of topology structure and graph labellings, and its difficulty is based on some mathematical problems such as graph isomorphism and graceful tree conjecture. Yao et al. [7, 8] proposed a new type of matrix combined with graph labelling technology to generate complex forms and longer byte text passwords. Tian et al. [9] proved that the password generated by the labelling of the graph can resist brute force attacks, copy attacks, and other attacks.
Graph labellings including the term of were used in many problems. An example, studied first by Graham and Sloane, is the harmonious graphs of modular versions of additive base problems stemming from error-correcting codes in the paper [10] in which the authors collected over 600 papers on various graph labellings. In 1981, Chang, Hsu, and Rogers [11] defined a strongly c-elegant labelling of a -graph . Shee [12] has proved that is strongly c-elegant for a particular value of and obtained several more specialized results pertaining to graphs formed from complete bipartite graphs. Seoud and Elsakhawi [13] have shown that with an edge joining two vertices of the same partite set is strongly c-elegant for . The strongly c-elegant labelling and some magic labelling involved in this paper all exist in the form of for edge in a graph.
The security of topological cryptography can be increased by using larger-scale graphs or richer graph coloring; therefore, it is worth studying to find the connections or conversion methods between graph labellings and how to construct large-scale graphs. In this paper, we study the relationships between strongly c-elegant labelling, super-edge magic total labelling, edge antimagic total labelling, and super -magical labelling; we also give a necessary condition for a graph with a strongly c-elegant labelling. Based on small-scale graphs with strongly c-elegant labelling, several kinds of large-scale graphs are constructed and the existence of their strongly c-elegant labelling is proved. Also, we present a transformation method for several strongly c-elegant labellings of a graph.
2. Preliminary
We list the symbols commonly used in this paper in Table 1; let and denote the vertex set and edge set of a graph , and other symbols can be found in [14, 15].
Definition 1. (see [11]). A mapping : such that the label of each edge in is defined by , and the resultant edge labels are distinct. If andwhere is a nonnegative integer, we say to be a strongly c-elegant labelling of ; then, the graph is called strongly c-elegant. When is a bipartite graph, that is, and , if for all and , we call a set-ordered strongly c-elegant labelling of (see an example in Figure 1(a)).
In 1970, Kotzig and Rosa defined the concept of edge magic total labelling (abbreviated as EMT labelling) of a -graph as follows.
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Definition 2. (see [16]). A bijective mapping from to such that for any edge ,where is a fixed constant, called a magic constant. Enomoto, Llad , Nakamigawa, and Ringel [17] called a graph with an edge magic total labelling that has the super-edge magic total labelling (SEMT labelling), and an example is shown in Figure 1(b).
Definition 3. (1)(see [10]). A -edge antimagic total labelling (abbreviated as -EAT labelling) is a bijection : having the setfor certain integers and ; analogously, a -EAT labelling of is super if .(2)A bijection of a -graph from to is regarded as a -magical labelling of if and only iffor integers and . Especially, if for any vertex , we say that is a super -magical labelling of (see Figure 1(c)).
Definition 4. (see (4)). Let : . The dual labelling of is defined by
3. Main Results
In this section, we prove that the strongly c-elegant labelling and SEMT labelling are pairwise equivalents via some super -magical labellings. There is a way of getting some graphs having EMT labellings from some strongly c-elegant graphs. Then, we show that a -graph admits a strongly c-elegant labelling and .
3.1. The Elegant Labelling and Magic Labelling
In order to achieve the purpose of mutual conversion between labellings, we firstly give the relationship between strongly c-elegant labelling and super-edge magic total labelling and we will show that the problem of determining those trees having EMT labellings can be reduced to the problem of finding strongly c-elegant labellings.
Theorem 1. A graph admits a strongly c-elegant labelling if and only if has SEMT labelling of .
The proof of Theorem 1 can be divided into the proofs of Lemmas 2 and 3.
Lemma 2. A -graph admits a strongly c-elegant labelling if and only if corresponds a super -magical labelling of for some integer .
Proof. Let be a strongly c-elegant labelling of , so we have and for an integer . We define another labelling of as for and if with . Let , and we haveWe notice that , and it shows that is a super -magical labelling of .
Conversely, assume that is a super -magical labelling of . We have and . Similar to the way above, we define a strongly c-elegant labelling of by for . Thereby, we define the label of each edge in as the following form:such that and where , as desired.
Lemma 3. A super -magical labelling of a graph can induce SEMT labelling of .
Proof. Let be a super -magical labelling of . Hence, , . According to the definition of -magical labelling, we havewhich gives us thatwhere ; it turns out that is a super -EAT labelling of . Hence, and . Since is a super -magical labelling of by Lemma 3, and there are for any edge , where .
We define another labelling of as for and for . Thereby, we havewhich implies that is SEMT labelling of since is a constant.
3.2. Some Properties on Elegant Labellings
Theorem 4. A connected -graph admits a strongly c-elegant labelling; then, . If is a set-ordered strongly c-elegant, then .
Proof. Since a strongly c-elegant labelling of with , where , we have a matrix as follows: if and if . Clearly, any entry is an element of the set . Thereby, we have .
Assume that is a strongly c-elegant labelling of , so that is bipartite and with and . We can define a matrix by when and ; otherwise, . Therefore, the number of different entries is at most equal to , which means since is connected.
Corollary 5. Let be a -regular -graph with ; if admits a strongly c-elegant labelling, then . Especially, is odd when and is even if .
Proof. Let be a strongly c-elegant labelling of the -regular -graph and . We computeand since . By Theorem 4, we have , which gives . When , we have ; thus, . For , we have and then .
4. Constructing Large-Scale Trees
Next, we construct several classes of large-scale trees having strongly c-elegant labellings through graph operations that connect edges between two vertices or overlap two vertices to form a new vertex.
Lemma 6. A tree on vertices admits a set-ordered strongly c-elegant labelling with ; then, where and , as well as for and The dual labelling of is a set-ordered strongly c-elegant labelling of with , where
Proof. Let be a set-ordered strongly c-elegant labelling with and for and ; therefore, . Since there are two vertices and such that , , and , we have that . If , we are done. Otherwise, , so there is an edge of with for such that , and moreover, and , which means that , a contradiction. Since , thenwhere .
Lemma 7. Given two trees on vertices with and , each admits a set-ordered strongly c-elegant labelling for .(1)Adjoin a certain vertex in to a certain vertex in such that the resultant graph admits a set-ordered strongly c-elegant labelling with (2)Identity a certain vertex in to a certain vertex in such that the resultant graph admits a set-ordered strongly c-elegant labelling with
Proof. Let for , by Lemma 6. Since when and for , there are for and for and for and for :(1)We have a tree obtained by adjoining of to of with an edge , and we give a set-ordered strongly c-elegant labelling as follows: if ; if ; if ; if . For an edge of , we notice thatThus, . It turns out that , where . If edge , and , and , there isso . We set and ; then, is a vertex partition of , which proves that is a set-ordered strongly c-elegant labelling of the graph .(2)We can obtain another tree by identifying vertex of to vertex of into a vertex. Next, we give a set-ordered strongly c-elegant labelling of as follows: if ; if ; if ; if .For , we have and . If , then , which leads that . is a vertex partition of , and
Theorem 8. Given two trees on vertices with and , each has a strongly c-elegant labelling with for . If is a set-ordered strongly c-elegant labelling and if there is an edge of with such that two components and of satisfy that(1) and , where .(2)Since are bipartite, and for ; there is , where , , , and and .
Then, there are two vertices and of such that the graph obtained by adjoining to with the edge and adjoining to with the edge between and and then deleting the edge has a strongly c-elegant labelling with .
Proof. Let be a set-ordered strongly c-elegant labelling of the bipartite graph on vertices from , such that for and and .
Therefore, there are two vertices and of such that for . We construct the graph by adjoining and between and and then deleting the edge . Clearly, is a tree with vertices and edges. Consequently, we define another labelling of as follows: if ; here, ; if ; if ; if ; and if .
Obviously, . In the following discussion, we shall estimate under this labelling .
Case 1. For , there is and . Thereby, we haveHence, .
Case 2. .
Case 3. For an edge with respect to and , we can obtain sinceand by Lemma 6.
Case 4. .
Case 5. For , there is and ; thereby, we haveIt turns out that .
We have shown that through Case 1 to Case 5, namely, is a strongly c-elegant labelling of .
An example illustrating Theorem 8 is shown in Figure 2.
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4.1. Generalized Caterpillars
Theorem 9. A caterpillar with vertices admits a set-ordered strongly c-elegant labelling.
Proof. Let be the set of all leaves of a caterpillar with vertices. By the definition of caterpillars, the graph is a path for . Let be the set of all leaves adjacent to with . Clearly, .
If , the caterpillar becomes a star; we have a set-ordered strongly c-elegant labelling of as follows: and for and for .
If , we define a set-ordered strongly c-elegant labelling of the caterpillar as follows. Let and . We definewith respect to and where if is even and if is odd.(1)For even integer , we let and label for , andwith respect to and . Thereby, we label . We set and .(2)For odd integer , we let and set for ,with respect to and ; therefore, .
The rest part of the proof is to verify the edge labels of . If the path has the even number of vertices, whereWhen is odd, there is , whereIn this case, we set and , so is vertex bipartite of caterpillar and ; therefore, we show is a set-ordered strongly c-elegant labelling of .
Adjoin a vertex of tree and a vertex of tree with an edge for ; the resulting tree is denoted as .
Theorem 10. If is a caterpillar for , then admits a set-ordered strongly c-elegant labelling.
Proof. Each tree with vertices and edges has a set-ordered strongly c-elegant labelling such that for and where and for . Let and , and we define a labelling on these trees by relabelling each vertex of , when : if ; if .
Generally, for , we have if and if .
If and and , then , so . We compute for , where , and we obtain , so .
We use to represent the maximum vertex label in and is the minimum vertex label in . We get graph by connecting vertices and , and the label of edge is ; obviously, for and . In addition, we set and , so is a vertex bipartite of tree ; it can be calculated that the largest label of the edge in is . Based on the above content, we obtain is a set-ordered strongly c-elegant labelling of (see an example in Figure 3).
4.2. Generalized Lobsters
In , when each tree is a star , then we have known that each lobster admits a set-ordered strongly c-elegant labelling and the symbol denotes . A spider is a tree whose vertices have at most degree 2, except only one vertex whose degree is not less than 3, and is called the center of the spider. Thereby, we can depict this spider by linking to an initial vertex of a path (called a leg of the spider) with vertices, , and write this spider by ; each indicates the length of th leg. If , abbreviate it as and name it in term of -regular spider. We will use some particular spiders to construct some large scale of lobsters.
Corollary 11. Lobster admit a set-ordered strongly c-elegant labelling.
For , let denote a spider with the center vertices and and legs , where is the initial vertex of the leg for . A spider-lobster is obtained by adjoining each pair of center vertices and between spiders and with an edge for . We say and the initial vertex and terminal vertex of the spider-lobster . If any is equal to for and , we denote this spider-lobster by . Moreover, if for in , thus we use a simple notation to replace .
Lemma 12. Each lobster has a strongly c-elegant labelling.
Proof. The number of vertices of lobster is , and we have the th leg of each spider for . We define a labelling of the lobster as follows: and for ; further, and for . Again, we set and . Thereby, since and . Furthermore, we have ; ; ; ; and , so , and it is easy to see that is a strongly c-elegant labelling of the lobster .
In spider-lobsters , we allow some spider to be an isolated vertex. We will find some strongly c-elegant labellings for two particular classes of spider-lobsters , . Let be even and be a positive integer. The first class of lobsters can be defined by setting if is odd; otherwise, in . Similarly, the second class of lobsters is constructed by letting if is odd; otherwise, in .
Lemma 13. Both lobsters and have a set-ordered strongly c-elegant labelling.
Proof. Notice that is a lobster on vertices, where and . Hence, for , we have where for , and when , we have . A labelling of the lobster can be defined in following forms:(1), , , and (2) and for (3) and for It is not hard to obtain and ; we set and , and , and it shows that is truly a set-ordered strongly c-elegant labelling of .
We have known the lobster where and . When , is the same as that of . For , . We construct a set-ordered strongly c-elegant labelling for the lobster as follows:(1), , , ;(2), , , and for (3) and for Notice that has vertices, and we can easily compute and ; we set , , and , so is a set-ordered strongly c-elegant labelling of .
Theorem 14. Suppose lobsters , where or 1 and . A lobster is determined by adjoining successively the initial vertex of to the terminal vertex of for ; this admits a set-ordered strongly c-elegant labelling.
Proof. Since admits a set-ordered strongly c-elegant labelling shown as in the proof of Lemma 13, we have and for all and for .
Let for . Adjoining the terminal vertex with to the initial vertex with , we have a lobster . We define a set-ordered strongly c-elegant labelling of as whenever ; if ; whenever ; and if . By the induction, the rest part of the proof of this theorem is obvious and we omit it.
Corollary 15. Suppose two lobsters having set-ordered strongly c-elegant labellings. A lobster constructed by adjoining the terminal vertex of a lobster to the initial vertex of the rest lobster has a set-ordered strongly c-elegant labelling.
4.3. Transformation on Strongly c-Elegant Labellings
Let be a strongly c-elegant labelling of a tree with vertices such that . Obviously, for an edge , the graph obtained by adding the edge to contains the uniquely cycle . We have three concepts in the following:(1)If there is an edge in such that , another tree admits this strongly c-elegant labelling and keeps . It is convenient to say as a left transformation of under and write . We are easy to define the inverse transformation of a left transformation , that is, by under .(2)Analogously, for , the graph has two cycles and such that and where and and . Thereby, we have a tree which admits this strongly c-elegant labelling and . We say to be a right transformation of under , denoted by . Clearly, where is the inverse transformation of a right transformation .(3)If there is an edge in such that and , the tree admits this strongly c-elegant labelling and keeps . We say as an increasing transformation of under and write . Moreover, we have . However, a star denies left and right transformations since the star on vertices has a uniquely strongly c-elegant labelling with . Thereby, we have a caterpillar that admits a strongly c-elegant labelling with ; namely, is an increasing transformation of the star under .
An example illustrating the above definition is shown in Figure 4.
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Theorem 16. Let be a tree with vertices having a strongly c-elegant labelling. If is not a star, either there is a left transformation of or a right transformation of .
Proof. Let be a strongly c-elegant labelling of a tree on vertices and . There are two vertices and such that . We take a vertex with since and is not a star. Thereby, the graph contains a cycle and there is an edge such that since .
If edge belongs to , we have a tree having the strongly c-elegant labelling with . Obviously, is a left transformation of .
If , we have , and furthermore, we can find a vertex in such that . Hence, the graph has two cycles. We have a right transformation of as admitted as a strongly c-elegant labelling with . This theorem has been proved.
5. Conclusion and Problems
In this paper, we mainly have studied the equivalence between strong c-elegant labelling, super-edge magic total labelling, edge antimagic total labelling, and super -magical labelling. Starting from small-scale graphs with strong c-elegant labelling, several large-scale graphs with strong c-elegant labellings are constructed. We have defined a transformation about strong c-elegant labelling. The object used in the encryption process is no longer just a labelling of a graph, but these methods can provide more text passwords for topological encryption and provide new ideas for expanding the topological password space. At the end of this paper, we propose several problems:
Problem 1. Can a tree that admits a strongly c-elegant labelling be transformed to a caterpillar ? That is to say, there is under . In other words, for a given tree , can we find a caterpillar such that , where and stand for the inverse transformations of and under , respectively?
Problem 2. Let be a set of all connected -graphs with the vertex set in common. If there is a labelling defined as for such that each graph of admits being as a strongly c-elegant itself, we say is a strongly c-elegant labelling of . For any , whether is there another graph such that where , , and ?
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Key Research and Development Plan under Grant no. 2019YFA0706401, the National Natural Science Foundation of China under Grants nos. 61632002, 61872166, and 61662066, and the National Natural Science Foundation of China Youth Project under Grant nos. 61902005 and 62002002.