Using Chinese Characters To Generate Text-Based Passwords For Information Security

07/11/2019 ∙ by Bing Yao, et al. ∙ 0

Graphical passwords (GPWs) are in many areas of the current world. Topological graphic passwords (Topsnut-gpws) are a new type of cryptography, and they differ from the existing GPWs. A Topsnut-gpw consists of two parts: one is a topological structure (graph), and one is a set of discrete elements (a graph labelling, or coloring), the topological structure connects these discrete elements together to form an interesting "story". Our idea is to transform Chinese characters into computer and electronic equipments with touch screen by speaking, writing and keyboard for forming Hanzi-graphs and Hanzi-gpws. We will use Hanzigpws to produce text-based passwords (TB-paws). We will introduce flawed graph labellings on disconnected Hanzi-graphs.

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I Introduction and preliminary

Security of cyber and information is facing more challenges and thorny problems in today’s world. There may exist such situation: a protection by the virtue of AI (artificial intelligence) resists attackers equipped by AI in current networks. We have to consider: How to overcome various attacker equipped by AI’s tools?

The origin of AI was generally acknowledged in Dartmouth Conference in 1956. In popularly, AI is defined as: “

Artificial intelligence (AI) is a branch of computer science. It attempts to understand the essence of intelligence and produce a new kind of intelligence machine that can respond in a similar way to human intelligence. Research in this field includes robots, language recognition, image recognition, natural language processing and expert systems

.”

In fact, the modern AI can be divided into two parts, namely, “artificial” and “intelligence”. It is difficult for computer to learn “qualitative change independent of quality” in terms of learning and “practice”. It is difficult for them to go directly from on “quality” to another “quality” or from one “concept” to another “concept”. Because of this, practice here is not the same practice as human beings, since the process of human practice includes both experience and creation.

For the above statement on AI, we cite an important sentence: A key signature of human intelligence is the ability to make ‘infinite use of finite means’ ([5] in 1836; [6] in 1965), as the beginning of an article entitled “

Relational inductive biases, deep learning, and graph networks

” by Battaglia et al. in [7]. They have pointed out: “in which a small set of elements (such as words) can be productively composed in limitless ways (such as into new sentences)”, and they argued that combinatorial generalization must be a top priority for AI to achieve human-like abilities, and that structured representations and computations are key to realizing this object. As an example of supporting ‘infinite use of finite means’, self-similarity is common phenomena between a part of a complex system and the whole of the system.

Yao et al. in [24] have listed some advantages of Chinese characters. Wang Lei, a teacher and researcher of Shenyang Institute of Education, stepped on the stage of “I am a speaker” and explained the beauty of Chinese characters as: (1) Chinese characters are pictographs, and each one of Chinese characters represents a meaning, a truth, a culture, a spirit. (2) Chinese characters are naturally topological structures. (3) The biggest advantage of Chinese characters is that the information density is very high. (4) Chinese characters is their inheritance and stability. Chinese characters are picturesque in shape, beautiful in sound and complete in meaning. It is concise, efficient and vivid, and moreover it is the most advanced written language in the world.

I-a Researching background

The existing graphical passwords (GPWs) were investigated for a long time (Ref. [14, 15, 16]). As an alternation, Wang et al. in [18] and [19] present a new-type of graphical passwords, called topological graphic passwords

(Topsnut-gpws), and show their Topsnut-gpws differing from the existing GPWs. A Topsnut-gpw consists of two parts: one is a topological structure (graph), and one is a set of discrete elements (here, a graph labelling, or a coloring), the topological structure connects these discrete elements together to form an interesting “story” for easily remembering. Graphs of graph theory are ubiquitous in the real world, representing objects and their relationships such as social networks, e-commerce networks, biology networks and traffic networks and many areas of science such as Deep Learning, Graph Neural Network, Graph Networks (Ref.

[7] and [8]). Topsnut-gpws based on techniques of graph theory, in recent years, have been investigated fast and produce abundant fruits (Ref. [43, 44, 46]).

As examples, two Topsnut-gpws is shown in Fig.1 (b) and (c).


Fig. 1: (a) A simplified Chinese character defined in [42]; (b) a mathematical model of ; (c) another mathematical model of .

Fig. 2: (a) A popular matrix of ; (b) the Hanzi-matrix of .

There are many advantages of Topsnut-gpws, such as, the space of Topsnut-gpws is large enough such that the decrypting Topsnut-gpws will be terrible and horrible if using current computer. In graph theory, Cayley’s formula (Ref. [10])

(1)

pointed that the number of spanning trees (tree-like Topsnut-gpws) of a complete graph (network) is non-polynomial, so Topsnut-gpws are computationally security; Topsnut-gpws are suitable for people who need not learn new rules and are allowed to use their private knowledge in making Topsnut-gpws for the sake of remembering easily; Topsnut-gpws, very often, run fast in communication networks because they are saved in computer by popular matrices rather than pictures; Topsnut-gpws are suitable for using mobile equipments with touch screen and speech recognition; Topsnut-gpws can generate quickly text-based passwords (TB-paws) with bytes as long as desired, but these TB-paws can not reconstruct the original Topsnut-gpws, namely, it is irreversible; many mathematical conjectures (NP-problems) are related with Topsnut-gpws such that they are really provable security.

The idea of “translating Chinese characters into Topsnut-gpws” was first proposed in [24]. Topsnut-gpws were made by Hanzi-graphs are called Hanzi-gpws (Ref. [34, 35, 36]), see a Hanzi-graph and a Hanzi-gpw are shown in Fig.1 (b) and (c). By the narrowed line under the Hanzi-matrix shown in Fig.2(a), we get a text-based password (TB-paw) as follows

and furthermore we obtain another TB-paw

along the narrowed line under the Hanzi-matrix shown in Fig.2(b). There are efficient algorithms for writing and from the Hanzi-matrices. It is not difficult to see there are at least TB-paws made by two matrices and , respectively.

There are many unsolved problems in graph theory, which can persuade people to believe that Topsnut-gpws can withdraw cipher’s attackers, such a famous example is: “If a graph with the maximum vertices has no a complete graph of vertices and an independent set of vertices, then we call a Ramsey graph and a Ramsey number. As known, it is a terrible job for computer to find Ramsey number , although we have known ”. Joel Spencer said:“Erdös asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for . In that case, he believes, we should attempt to destroy the aliens”.

I-B Researching tasks

Although Yao et al. [24] have proposed Hanzi-graphs and Hanzi-gpws, however, we think that their junior work is just a beginning on Topsnut-gpws made by the idea of “Hanzi-graphs puls graph labellings”.

Our goals are: (1) To design passwords of Chinese characters by voice inputting, hand inputting into computers and mobile equipments with touch screen; (2) to make more complex TB-paws for encrypting electronic files, or encrypting networks.

In technique, we will introduce how to construct mathematical models of Chinese characters, called Hanzi-graphs, and then use Hanzi-graphs and graph labelling/colorings to build up Hanzi-graph passwords, called Hanzi-gpws. Then, several types of Hanzi-matrices will be defined for producing TB-paws. Moreover, we will explore to encrypt dynamic networks, such as deterministic networks, scale-free networks, self-similar networks, and so on.

In producing TB-paws from Hanzi-gpws, we can get TB-paws with hundreds bytes. As known, brute-force attacks work by calculating every possible combination that could make up a password and testing it to see if it is the correct password. As the password’s length increases, the amount of time, on average, to find the correct password increases exponentially. AES (Advanced Encryption Standard) permits the use of 256-bit keys. How many possible combinations of (or 256-bit) encryption are there? There are

115,792,089,237,316,195,423,570,985,008,687,907,853,

269,984,665,640,564,039,457,584,007,913,129,639,936
(78 digits) possible combinations for 256-bit keys [1]. Breaking a symmetric 256-bit key by brute force requires times more computational power than a 128-bit key. Fifty supercomputers that could check a billion billion () AES keys per second (if such a device could ever be made) would, in theory, require about years to exhaust the 256-bit key space, cited from “Brute-force attack” in Wikipedia.

I-C Preliminaries: terminology, notation and definitions

Undefined labelling definitions, terminology and algorithms mentioned here are cited from [10] and [11]. The following terminology and notation will be used in this article:

  1. Hanzis (Chinese characters) mentioned here are listed in GB2312-80 encoding of Chinese characters, in which there are 6763 simplified Chinese characters and 682 signs (another Chinese encoding is GBK, formed in Oct. 1995, containing 21003 simplified Chinese characters and 883 signs, [42]).

  2. A -graph has vertices (nodes) and edges (links), notations and are the sets of vertices and edges of , respectively.

  3. The number of elements of a set is called cardinality, denoted as .

  4. The set of neighbors of a vertex is denoted as , and the number of elements of the set is denoted as , also, is called the degree of the vertex , very often, write .

  5. A vertex is called a “leaf” if its degree .

  6. A subgraph of a graph is called a vertex-induced subgraph over a subset of if and for any . Very often, we write this subgraph as .

  7. An edge-induced graph over an edge subset of is a subgraph having its edge set and its vertex set containing two ends of every edge of .

We will use various labelling techniques of graph theory in this article.

Definition 1.

[26] A labelling of a graph is a mapping such that for any pair of elements of , and write the label set . A dual labelling of a labelling is defined as: for . Moreover, is called the vertex label set if , the edge label set if , and the universal label set if . Furthermore, if is a bipartite graph with vertex bipartition , and holds , we call a set-ordered labelling of .

We use a notation to denote the set of all subsets of a set . For instance, , so has its own elements: , , , , , and . The empty set is not allowed to belong to hereafter. We will use set-type of labellings defined in the following Definition 2.

Definition 2.

[26] Let be a -graph . We have:

(i) A set mapping is called a total set-labelling of if for distinct elements .

(ii) A vertex set mapping is called a vertex set-labelling of if for distinct vertices .

(iii) An edge set mapping is called an edge set-labelling of if for distinct edges .

(iv) A vertex set mapping and a proper edge mapping are called a v-set e-proper labelling of if for distinct vertices and two edge labels for distinct edges .

(v) An edge set mapping and a proper vertex mapping are called an e-set v-proper labelling of if for distinct edges and two vertex labels for distinct vertices .

Ii Translating Chinese characters into graphs

Hanzis, also Chinese characters, are expressed in many forms, such as: font, calligraphy, traditional Chinese characters, simplified Chinese characters, brush word, etc. As known, China Online Dictionary includes Xinhua Dictionary, Modern Chinese Dictionary, Modern Idiom Dictionary, Ancient Chinese Dictionary, and other 12 dictionaries total, China Online Dictionary contains about 20950 Chinese characters; 520,000 words; 360,000 words (28,770 commonly used words); 31920 idioms; 4320 synonyms; 7690 antonyms; 14000 allegorical sayings; 28070 riddles; and famous aphorism 19420.

Ii-a Two types of Chinese characters

In general, there are two type of Chinese characters used in the world, one is called traditional Chinese characters and another one simplified Chinese characters, see Fig.3. We, very often, call a traditional Chinese characters or a simplified Chinese characters as a Hanzi (Chinese character).

The stroke number of a Hanzi is less than that of the traditional Chinese character corresponding with , in general. We can compute the difference of two strokes of two-type Chinese characters and , denoted as . For example, , where the Hanzi is shown in Fig.3(13). And, , where the Hanzi is shown in Fig.3(3).


Fig. 3: Left is a simplified Chinese character, and Right is a traditional Chinese character in each of equations above.

Some Hanzis are no distinguishing about traditional Chinese characters and simplified Chinese characters.


Fig. 4: Three Chinese characters with more strokes, Left has 64 strokes, Middle has 56 strokes.

Ii-B Different fonts of Hanzis

There are four fonts in printed Hanzis. In Fig.5, we give four basic fonts: Songti, Fangsong, Heiti and Kaiti. Clearly, there are differences in some printed Hanzis. These differences will be important for us when we build up mathematical models of Hanzis.


Fig. 5: Difference between four fonts in printed Hanzis.

Ii-C Matching behaviors of Hanzi-graphs


Ii-C1 Dui-lians, also, Chinese couplets

In Chinese culture, a sentence, called “Shang-lian”, has its own matching sentence, named as “Xia-lian”, and two sentences Shang-lian and Xia-lian form a Chinese couplet, refereed as “Dui-lian” in Chinese. The sentence (a) of Fig.6 is a Shang-lian, and the sentence (b) of Fig.6 is a Xia-lian of the Shang-lian (a). We can use Dui-lians to design Topsnut-gpws. For example, we can consider the Shang-lian (a) shown in Fig.6 as a public key, the Xia-lian (b) shown in Fig.6 as a private key, and the Dui-lian (c) as the authentication. Moreover, the Dui-lian (c) can be made as a public key, and it has its own matching Dui-lian (d) as a private key, we have the authentication (e) of two Dui-lians (c) and (d). However, Dui-lians have their complex, for instance, the Shang-lian (f) shown in Fig.6 has over candidate private keys. As known, a Dui-lian “Chongqing Yonglian” written by Xueyi Long has 1810 Hanzis. Other particular Chinese couplets are shown in Fig.7 and Fig.8.


Fig. 6: Couplets: (a) is a public key; (b) is a private key; (c) is the authentication of the public key (a) and the private key (b); (d) is a private key matching with the public key (c); (e) is the authentication of (c) and (d); (f) is a famous public key having no matching, although there are over candidate private keys.

Fig. 7: One Hanzi (Chinese character) may appear two or more times in a Dui-lian (couplet).

Fig. 8: The same Hanzis in a couplet.

Ii-C2 Conundrums in Chinese

Chinese riddles (also “Miyu”) are welcomed by Chinese people, and Chinese riddles appear in many where and actions of China. (see Fig. 9)


Fig. 9: Twelves Chinese conundrums.

Ii-C3 Chinese Xie-hou-yu

Xie-hou-yu” is a two-part allegorical saying, of which the first part, always stated, is descriptive, while the second part, sometimes unstated, carries the message (see Fig.10).


Fig. 10: Some Xie-hou-yus in Chinese.

Ii-C4 Chinese tongue twisters

Chinese tongue twisters are often applied in Chinese comic dialogue (cross talk), which are popular in China. (see Fig. 11)


Fig. 11: Three Chinese tongue twisters.

Ii-C5 Understanding by insight, homonyms

Such examples shown in Fig.12.


Fig. 12: Private keys obtained by homonyms, or understanding by insight.

Ii-C6 Same pronunciation, same Pianpang

In Fig.13, we can see eight Hanzis with the same pronunciation shown in Fig.13 (a) and ten Hanzis with the same Pianpang shown in Fig.13 (b). Moreover, all Hanzia have the same pronunciation “ji” in a famous Chinese paragraph shown in Fig.14.


Fig. 13: (a) Ten Hanzis with the same pronunciation; (b) twelves Hanzis with the same Pianpang.

Fig. 14: All Hanzia have the same pronunciation “ji”.

Ii-C7 Chinese dialects

(also, “Fangyan”) One Chinese word may have different replacements in local dialects of Chinese. For example, father, daddy can be substituted as Fig.15. And, different expressions of a sentence “Daddy, where do you go?” is shown in Fig.16.


Fig. 15: Father, daddy in Chinese dialects.

Fig. 16: Daddy, where do you go?

Ii-C8 Split Hanzis, building Hanzis

An example is shown in Fig.17 (a) for illustrating “split a word into several words”, and Fig.17 (b) is for building words by a given word.


Fig. 17: (a) Split a word into several words; (b) building words by a group of words obtained from splitting a given word.

Ii-C9 Explaining Hanzis

See examples are shown in Fig.18.


Fig. 18: Explaining words.

Ii-C10 Tang poems

As known, there are at least 5880195 Tang poems in China (see Fig.19).


Fig. 19: Tang poems.

Ii-C11 Idioms and Hanzi idiom-graphs

A Hanzi idiom-graph (see Fig.20) is one labelled with Hanzi idioms by a vertex labelling , two vertices are joined by an edge labelled with .


Fig. 20: A Hanzi idiom-graph.

Ii-C12 Traditional Chinese characters are complex than Simplified Chinese characters

Expect the stroke number of a traditional Chinese character is greater than that of a simplified Chinese character, the usage of some traditional Chinese characters, also, is not unique, such examples are shown in Fig.21.


Fig. 21: The usage of a traditional Chinese character B with is not unique.

Ii-C13 Configuration in Hanzis

  1. Symmetry means that Hanzis posses horizontal symmetrical structures, or vertical symmetrical structures, or two directional symmetries. We select some Hanzis having symmetrical structures in Fig.22 (a), (b), (c) and (f).

  2. Overlapping Hanzis. See some overlapping Hanzis shown in Fig.22 (d), (e), (f) and (g). Moreover, in Fig.22 (g), a Hanzi (read ‘shuāng’) (2-overlapping Hanzi) is consisted of two Hanzis (read ‘yòu’), and another (read ‘ruò’) (3-overlapping Hanzi) is consisted of three Hanzis . Moreover, four Hanzi construct a Hanzi (read as ‘zhuó’, 4-overlapping Hanzi).


Fig. 22: Hanzis with symmetrical structure and overlapping structure.

Ii-D Mathematical models of Hanzis

We will build up mathematical models of Hanzis, called Hanzi-graphs, in this subsection.


Ii-D1 The existing expressions of Hanzis

In fact, a Hanzi has been expressed in the way: (1) a “pinyin” in oral communication, for example, the pinyin “rén” means “man”, but it also stands for other 12 Hanzis at least (see Fig.23(a)); (2) a word with four English letters and numbers of , for instance, “rén”=4EBA (see Fig.23(b), also called a code); (3) a number code “4043” defined in “GB2312-80 Encoding of Chinese characters” [42]S, which is constituted by (see Fig.23(c)).

Clearly, the above three ways are not possible for making passwords with bytes as long as desired. We introduce the fourth way, named as Topsnut-gpw, see an example shown in Fig.1(c).


Fig. 23: Four expressions of a Hanzi (= man).

As known, Hanzi-graphs are saved in computer by popular matrices, see a Hanzi-graph shown in Fig.1 (b) and its matrix shown in Fig.2 (a).


Fig. 24: Three substituted expressions of eight Hanzis.

Fig. 25: The meaningful paragraphs obtained from nine Hanzis of GB2312-80.

In Fig.25, we use two expressions (a1) and (a2) to substitute a Chinese sentence (a), that is, (a)=(a1), or (a)=(a2). By this method, we have

(a1) ;

(a2) .

(b1) ;

(b2)

(c1) ;

(c2)

(d1) ;

(d2)

(e1) ;

(e2)

(f1) ;

(f2)

(g1) ;

(g2)

(h1) ;

(h2)

(i1) ;

(i2)

(j1) ;

(j2)

(k1) ;

(k2)

(l1) ;

(l2)

Fig.25 shows some permutations of nine Hanzis , , , , , , , , . In fact, there are about permutations made by these nine Hanzis. If a paragraph was made by a fixed group of 50 Hanzis, then we may have about paragraphs made by the same group . So, we have enough large space of Hanzi-graphs for making Hanzi-gpws.


Ii-D2 Basic rules for Hanzi-graphs

For the task of building mathematical models of Hanzis, called Hanzi-graphs, we give some rules for transforming Hanzis into Hanzi-graphs.

  1. Stroke rule. It is divided into several parts by the strokes of Hanzis. Some examples are shown in Fig.26 and Fig.27 based on “Pianpang” of Hanzis.

  2. Crossing and overlapping rules. Hanzi-graphs are obtained by the crossing and overlapping rules (see Fig.27 (a) and Fig.28).


    Fig. 26: Hanzi-graphs with one stroke, in which Hanzi-graphs (b), (c) and (d) can be considered as one from the topology of view. So, (a) and (e) are the same Hanzi-graph.

    Fig. 27: Hanzi-graphs with two strokes. According to the topology of view, Hanzi-graphs (b), (c) and Hanzi-graphs (f) and (g) shown in Fig.26 can be considered as one; Hanzi-graphs (e), (f), (i) and Hanzi-graphs (h) shown in Fig.26 are the same; and Hanzi-graphs (g) and (h) are the same.

    Fig. 28: Hanzi-(a-) is transformed into Hanzi-graph-(b-) with .
  3. Set-orderedable rule. We abide for fonts: songti, fangsong, heiti and kaiti to construct Hanzi-graphs that admit set-ordered graceful labellings (see Definition 4). Some examples are shown in Fig.29 and Fig.30.

  4. No odd-cycles.

    We restrict our Hanzi-graphs have no odd-cycles for the guarantee of set-ordered graceful labellings (see Fig.

    30). There are over 6763 Hanzis in [42], and we have 3500 Hanzis in frequently used. So it is not an easy job to realize the set-ordered gracefulness of the Hanzi-graphs in [42]. Clearly, the 0-rotatable gracefulness of the Hanzi-graphs in [42] will be not slight, see Definition 35.


Fig. 29: First group of mathematical models of Hanzis components and radicals.

Fig. 30: Second group of mathematical models of Hanzis components and radicals.

A group of Hanzi-graphs made by Rule- with is shown in Fig.31. If a Hanzi-graph is disconnected, and has components, we refer to it as a -Hanzi-graph directly.


Fig. 31: The topological structure (Hanzi-graphs) of Hanzis shown in Fig.24.

Ii-E Space of Hanzi-graphs

A list of commonly used Hanzis in modern Chinese was issued by The State Language Work Committee and The State Education Commission in 1988, with a total of 3500 characters. The commonly used part of the Hanzis with a coverage rate of 97.97% is about 2500 characters. This means that the commonly used 2500 characters can help us to make a vast space of Hanzi-graphs.

For example, the probability of a Hanzi appearing just once in a Chinese paragraph is a half, so the space of paragraphs made in Hanzis contains at lest

elements, which is far more than the number of sands on the earth. It is known that the number of sands on the earth is about , or about

, someone estimates the number of sands on the earth as

.

Iii Mathematical techniques

Since some Topsnut-gpws were made by graph coloting/labellings, we show the following definitions of graph coloting/labellings for easily reading and quickly working.

Iii-a Known labellings

Definition 3.

[29] An edge-magic total graceful labelling of a -graph is defined as: such that for any two elements , and with a constant for each edge . Moreover, is super if (or ).

In Definition 4 we restate several known labellings that can be found in [11], [31], [48, 49] and [23]. We write and hereafter.

Definition 4.

Suppose that a connected -graph admits a mapping . For edges the induced edge labels are defined as . Write , . There are the following constraints:

  1. .

  2. .

  3. , .

  4. , .

  5. .

  6. .

  7. is a bipartite graph with the bipartition such that ( for short).

  8. is a tree containing a perfect matching such that for each edge .

  9. is a tree having a perfect matching such that for each edge .

Then we have: a graceful labelling satisfies (a), (c) and (e); a set-ordered graceful labelling holds (a), (c), (e) and (g) true; a strongly graceful labelling holds (a), (c), (e) and (h) true; a strongly set-ordered graceful labelling holds (a), (c), (e), (g) and (h) true. An odd-graceful labelling holds (a), (d) and (f) true; a set-ordered odd-graceful labelling holds (a), (d), (f) and (g) true; a strongly odd-graceful labelling holds (a), (d), (f) and (i) true; a strongly set-ordered odd-graceful labelling holds (a), (d), (f), (g) and (i) true.

Definition 5.

A total graceful labelling of a -graph is defined as: such that for each edge , and for any two elements . Moreover, is super if (or ).

Definition 6.

Let be a -graph having vertices and edges, and let for integers and .

(1) [11] A felicitous labelling of holds: , for distinct and ; and furthermore, is super if .

(2) [13] A -graceful labelling of holds , for distinct and . Especially, a -graceful labelling is also a -graceful labelling.

(3) [11] An edge-magic total labelling of holds such that for any edge , where the magic constant is a fixed integer; and furthermore is super if .

(4) [11] A -edge antimagic total labelling of holds and , and furthermore is super if .

(5) [49] An odd-elegant labelling of holds , for distinct , and .

(6) [9] A labeling of is said to be -arithmetic if , for distinct and .

(7) [11] A harmonious labelling of holds , and such that (i) if is not a tree, for distinct ; (ii) if is a tree, for distinct , and for some .

Definition 7.

[26] Let be a total labelling of a -graph . If there is a constant such that , and each edge corresponds another edge holding , then we name as a relaxed edge-magic total labelling (relaxed Emt-labelling) of (called a relaxed Emt-graph).

Definition 8.

[26] Suppose that a -graph admits a vertex labelling and an edge labelling . If there is a constant such that for each edge , and , then we refer to as an odd-edge-magic matching labelling (Oemm-labelling) of (called an Oemm-graph).

Definition 9.

[26] Suppose that a -graph admits a vertex labelling and an edge labelling , and let for . If (i) each edge corresponds an edge such that ; (ii) and there exists a constant such that each edge has a matching edge holding true; (iii) there exists a constant such that for each edge . Then we call an ee-difference odd-edge-magic matching labelling (Eedoemm-labelling) of (called a Eedoemm-graph).

Definition 10.

[26] A total labelling for a bipartite -graph is a bijection and holds:

(i) (e-magic) ;

(ii) (ee-difference) each edge matches with another edge holding (or );

(iii) (ee-balanced) let for , then there exists a constant such that each edge matches with another edge holding (or ) true;

(iv) (EV-ordered) (or , or , or , or is an odd-set and is an even-set);

(v) (ve-matching) there exists a constant such that each edge matches with one vertex such that , and each vertex matches with one edge such that , except the singularity ;

(vi) (set-ordered) (or ) for the bipartition of .

We refer to as a 6C-labelling.

Definition 11.

[26] Suppose that a -graph admits a vertex labelling and an edge labelling , and let for . If there are: (i) each edge corresponds an edge such that (or ); (ii) and there exists a constant such that each edge has a matching edge holding true; (iii) there exists a constant such that for each edge ; (iv) there exists a constant such that each edge matches with one vertex such that , and each vertex matches with one edge such that , except the singularity . Then we name as an ee-difference graceful-magic matching labelling (Dgemm-labelling) of (called a Dgemm-graph).

Definition 12.

[26] Let be a labelling of a -graph , and let

we say to be a difference-sum labelling. Find two extremum (profit) and (cost) over all difference-sum labellings of .

Definition 13.

[26] Let be a labelling of a -graph , and let

we call a felicitous-sum labelling. Find two extremum and over all felicitous-sum labellings of .

Motivated from Definition 12 and Definition 13, we design:

Definition 14.

A connected -graph admits a labelling , such that for any pair of elements . We have the following sums:

(2)
(3)

and

(4)

We call to be: (1) a ve-sum-difference labelling of if it holds (2) true; (2) a ve-difference labelling of if it holds (3) true; (3) a k-edge-average labelling of if it holds (4) true.

Find these six extremum , , , , and over all -labellings of , where ve-sum-difference, ve-difference, k-edge-average.

Definition 15.

[47] Let be the bipartition of a bipartite -graph . If admits a felicitous labelling such that , then we refer to as a set-ordered felicitous labelling and a set-ordered felicitous graph, and write this case as , and moreover is called an optimal set-ordered felicitous labelling if and .

Definition 16.

[27] A -graph admits an edge-odd-graceful total labelling and such that

with .

Definition 17.

[27] A -graph admits a multiple edge-meaning vertex labelling such that (1) and a constant ; (2) and a constant ; (3) and ; (4) and a constant ; (5) an odd number for each edge holding , and with .

Definition 18.

[27] A -graph admits a vertex set-labelling  (or , and induces an edge set-labelling . If we can select a representative for each edge label set with such that

we then call a graceful-intersection (or an odd-graceful-intersection) total set-labelling of .

Definition 19.

[27] Let be an every-zero graphic group. A -graph admits a graceful group-labelling (or an odd-graceful group-labelling) such that each edge is labelled by under a zero , and (or ).

Definition 20.

[27] Let be an odd-graceful labelling of a -graph , such that and . If , then is called a perfect odd-graceful labelling of .

Definition 21.

[27] Suppose that a -graph admits an -labelling . If , we call a perfect -labelling of .

Definition 22.

[27] Let be a labelling of a -graph and let each edge have its own label as with . If each edge holds true, where is a positive constant, we call and