mathematics

Article Matching-Type Image-Labelings of Trees

Jing Su 1,†, Hongyu Wang 2,† and Bing Yao 3,*,†

1 School of Electronics Engineering and Computer Science, Peking University, No. 5 Yiheyuan Road, Haidian Distruct, Beijing 100871, China; [email protected] 2 National Computer Network Emergency Response Technical Team/Coordination Center of China, Beijing 100029, China; [email protected] 3 College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China * Correspondence: [email protected]; Tel.: +86-1335-942-8694 † These authors contributed equally to this work.

Abstract: A variety of labelings on trees have emerged in order to attack the Graceful Tree Conjecture, but lack showing the connections between two labelings. In this paper, we propose two new labelings: vertex image-labeling and edge image-labeling, and combine new labelings to form matching-type image-labeling with multiple restrictions. The research starts from the set-ordered graceful labeling of the trees, and we give several generation methods and relationships for well-known labelings and two new labelings on trees.

Keywords: graph labeling; tree; image-labeling; graceful labeling



1. Introduction and Preliminary
 1.1. A Simple Introduction Citation: Su, J.; Wang, H.; Yao, B. A graph labeling is an assignment of non-negative integers to the vertices and edges Matching-Type Image-Labelings of of a graph subject to certain conditions. The problem of graph labeling can be traced to a Trees. Mathematics 2021, 9, 1393. well-known Ringel–Kotzig Decomposition Conjecture in popularization [1]: “A complete https://doi.org/10.3390/ graph K2n+1 can be decomposed into 2n + 1 subgraphs that are all isomorphic with a given tree math9121393 of n edges”. This conjecture is explained and traced by Kotzig to the series of attacks intent on proving that trees are graceful. Labeling as a technique is applied to X-ray Academic Editors: Sergey Kitaev, crystallography [2], communication network addressing [3], information encryption [4], Janez Žerovnik and Frank Werner radio channel assignment [5] and so on. In the past fifty years, starting from different

Received: 2 April 2021 practical problems, many types of new labelings have emerged. Gallian, in [6], distributes Accepted: 10 June 2021 a survey on graph labeling. The famous conjecture is as follows. Published: 16 June 2021 Conjecture 1 ([7]). Every tree is graceful (Graceful Tree Conjecture, GTC).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in GTC has aroused interest in graph labelings, and people have put forward a lot published maps and institutional affil- of labelings according to different practical problems. The odd-graceful labeling was iations. introduced by Gnanajothi in 1991, and she conjectured:

Conjecture 2 ([8]). All trees admit odd-graceful labellings (OGTC).

Zhou et al. have shown that “Every lobster is odd-graceful”[9]. In 1981, Chang, Hsu, Copyright: © 2021 by the authors. and Rogers [10] defined an elegant labeling f of a graph G with q edges as an injective Licensee MDPI, Basel, Switzerland. function from the vertices of G to the set {0, 1, 2, ··· , q} such that each edge xy is assigned This article is an open access article ( ) + ( ) ( + ) distributed under the terms and the label f x f y mod q 1 , and the resulting edge labels are nonzero and distinct. conditions of the Creative Commons Then, the odd-elegant labeling was developed on the basis of this labeling. In 1970, Kotzig Attribution (CC BY) license (https:// and Rosa defined the magic labeling of graphs. Inspired by Kotzig-Rosa notion, Enomoto, creativecommons.org/licenses/by/ Lladó, Nakamigawa and Ringel [11] called a graph G with an edge-magic total labeling 4.0/). that has the additional property that the vertex labels from 1 to |V| are super edge-magic

Mathematics 2021, 9, 1393. https://doi.org/10.3390/math9121393 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 1393 2 of 8

total labeling [12]. Acharya and Hegde [13] have generalized sequential labelings to (k, d)-arithmetic total labeling and (k, d)-graceful labeling with positive integers k and d. The algorithmic research on the labeling of graphs is interesting [14]. Among them, the research results on special graphs are the most significant. Gao et al. [15] showed relevant conclusions on the antimagic orientation of lobsters, and Sethuraman et al. [16] proved that any acyclic graph can be embedded in a unicyclic graceful graph. In this paper, inspired by graphical passwords, we put forward the concepts of new image-labelings: vertex image-labeling and edge image-labeling. We show the relationships between labelings on trees by producing several image-labelings from the set-ordered graceful labeling. Such results can be applied to molecular structures [17] and asymmetric cryptosystem [18], therefore, the theoretical research on labeling is meaningful.

1.2. Preliminary Graphs mentioned here are simple and undirected. Let G = (V, E) be a graph with vertex set V and edge set E, if the vertex set V can be divided into two disjoint subsets V1 and V2, such that the two end-vertices v1 and v2 of an edge v1v2 belong to two different vertex sets, that is v1 ∈ V1 and v2 ∈ V2, then G is called a bipartite graph. A (p, q)-graph is a graph with p vertices and q edges. The number of elements in a set X is written as |X|. We will use an integer set Sk,d = {k, k + d, k + 2d, ... , k + (q − 1)d} for integers k ≥ 0 and d ≥ 1, and use [x, y] to stand for an integer set {x, x + 1, ... , y} with two integers x, y subject to 0 ≤ x < y, as well as use [s, t]o to indicate an integer set {s, s + 2, ... , t} for two odd integers s, t holding 1 ≤ s < t. All numbers are integers, and other notations and terminologies not introduced here can be referred to [19]. Suppose that a (p, q)-graph G admits a mapping f : S → [a, b] with S ⊆ V(G) ∪ E(G), we write the label set { f (u) : u ∈ S} by f (S), and we restate several well-known labelings as follows:

Definition 1 ([20]). If the mapping θ holds θ(V(G)) ⊆ [0, q] with min θ(V(G)) = 0, θ(u) 6= θ(v) for distinct u, v ∈ V(G), and θ(E(G)) = {θ(uv) = |θ(u) − θ(v)| : uv ∈ E(G)} = [1, q], then we call θ a graceful labeling of G. If the (p, q)-graph G is a bipartite graph with vertex partition (X, Y), and a graceful labeling θ holds max{θ(u) : u ∈ X} < min{θ(v) : v ∈ Y} (abbreviated as max θ(X) < min θ(Y)), we call θ a set-ordered graceful labeling.

Definition 2 ([6]). A (p, q)-graph G admits a mapping α holding α(V(G)) = { f (u) : u ∈ V(G)} ⊆ [0, 2q − 1] and min α(V(G)) = 0, α(u) 6= α(v) for distinct vertices u, v ∈ V(G), α(E(G)) = {α(uv) = |α(u) − α(v)| : uv ∈ E(G)} = [1, 2q − 1]o, then α is called an odd- graceful labeling of graph G.

Definition 3 ([6]). If a (p, q)-graph G has a function β holds β(V(G)) ⊆ [0, 2q − 1], as well as β(u) 6= β(v) for distinct vertices u, v ∈ V(G), and β(E(G)) = {β(uv) = β(u) + β(v)(mod 2q) : uv ∈ E(G)} = [1, 2q − 1]o, we call β an odd-elegant labeling of graph G.

Definition 4 ([13]). If there is a labeling δ of a (p, q)-graph G holding δ(V(G)) ⊆ [0, k + (q − 1)d] such that δ(u) 6= δ(v) for distinct u, v ∈ V(G), δ(E(G)) = {δ(uv) = |δ(u) − δ(v)| : uv ∈ E(G)} = {k, k + d, k + 2d, ... , k + (q − 1)d}, then we call δ a (k, d)-graceful labeling of graph G, where k and d are positive integers.

Definition 5 ([6]). Let k and d be positive integers. A labeling γ of a (p, q)-graph G is said to be (k, d)-arithmetic total labeling if γ(V(G)) ⊆ [0, k + (q − 1)d], γ(u) 6= γ(v) for distinct vertices u, v ∈ V(G), and {γ(u) + γ(v) : uv ∈ E(G)} = {k, k + d, k + 2d,..., k + (q − 1)d} holds. Mathematics 2021, 9, 1393 3 of 8

Definition 6 ([6]). If the mapping σ holds σ(V(G)) ∪ σ(E(G)) = [1, p + q], and σ(u) + σ(v) + σ(uv) = C for each edge uv ∈ E(G), where C is a magic constant, then we call σ an edge-magic total labeling of G; moreover, σ is called a super edge-magic total labeling if σ(V(G)) = [1, p].

Next, we define the following new labelings to explore the relationships between several known labelings introduced here.

Definition 7. Let f : V(G) ∪ E(G) → [a, b] and g : V(G) ∪ E(G) → [a0, b0] be two labelings of a (p, q)-graph G, integers a, b, a0, b0 satisfy 0 ≤ a < b and 0 ≤ a0 < b0. (1) An equation f (v) + g(v) = k0 holds true for each vertex v ∈ V(G), where k0 is a positive constant and it is called vertex-image coefficient, then f and g are called a matching of vertex image-labelings (abbreviated as v-image-labelings); (2) An equation f (uv) + g(uv) = k00 holds true for every edge uv ∈ E(G), and k00 is a positive constant, called edge-image coefficient, then both labelings f and g are called a matching of edge image-labelings (abbreviated as e-image-labelings).

If labelings f and g are the same labeling functions, then they are called a matching of W-type v-image-labelings, or a matching of W-type e-image-labelings, where “W-type” ∈ {graceful, odd-graceful, odd-elegant, (k, d)-graceful, (k, d)-arithmetic total, super edge- magic total}. See an example in Figure1, f1 and g1 are a matching of set-ordered graceful v-image-labelings, f1 and h1 are a matching of set-ordered graceful e-image-labelings.

0 5 6 1 3 5 5 6 2 5 6 2 2 1 1 5 6 5 0 2 4 3 3 3 3 4 0 4 1 4 1 3 6 2 4 4 2 1 6 f1 g1 h1

(a) f11( v ) g ( v ) 6 (b) f11( uv ) h ( uv ) 7

Figure 1. A tree T admits: (a) a matching of set-ordered graceful v-image-labelings f1 and g1,(b) a matching of set-ordered graceful e-image-labelings f1 and h1. 2. Main Results and Proofs Theorem 1. Let T be a tree with p vertices and q edges, (X, Y) is the bipartition of vertices of T with |X| = s, |Y| = t and p = s + t. If the tree T admits a set-ordered graceful labeling, then it admits a matching of graceful v-image-labelings and a matching of graceful e-image-labelings.

Proof. Since any tree is bipartite, and (X, Y) is the bipartition of vertices of a tree T, where X = {xi : i ∈ [1, s]} and Y = {yj : j ∈ [1, t]} holding s + t = |V(T)| = p. Clearly, xi ∈ X and yj ∈ Y for each edge xiyj of T. By the hypothesis of the theorem, the tree T admits a set- ordered graceful labeling f , without loss of generality, we have f (xi) = i − 1 with i ∈ [1, s], f (yj) = s + j − 1 with j ∈ [1, t] and f (xiyj) = f (yj) − f (xi) = s + j − i for every edge xiyj ∈ E(T), obviously, condition max f (X) < min f (Y) is satisfied. We define another set-ordered graceful labeling g of T as follows: g(xi) = s + t − 1 − f (xi) with i ∈ [1, s], g(yj) = s + t − 1 − f (yj) with j ∈ [1, t], and g(xiyj) = f (xiyj) for every edge xiyj ∈ E(T). Therefore, we have f (v) + g(v) = s + t − 1 = p − 1 for every vertex v ∈ V(T). According to Definition1, f and g are a matching of graceful v-image-labelings with vertex-image coefficient k0 = p − 1. There is another labeling h of T defined as: h(xi) = s − 1 − f (xi) = s − i with i ∈ [1, s], h(yj) = t + 2s − 1 − f (yj) = t + s − j with j ∈ [1, t], since 0 ≤ s − i ≤ s − 1 and s ≤ t + s − j ≤ p − 1 = q, then max h(X) = s − 1 < s = min h(Y) and the vertex label set h(V(T)) = [0, q], in addition, h(xiyj) = |h(xi) − h(yj)| = h(yj) − h(xi) = t + i − j for Mathematics 2021, 9, 1393 4 of 8

each edge xiyj ∈ E(T) with 1 ≤ t + i − j ≤ q, so the edge label set h(E(T)) = [1, q], which shows that the labeling h is just a set-ordered graceful labeling of T. In addition, we find

h(xiyj) = h(yj) − h(xi) = t + 2s − 1 − f (yj) − [s − 1 − f (xi)] (1) = t + s − [ f (yj) − f (xi)] = t + s − f (xiyj)

for each edge xiyj ∈ E(T), immediately, we get f (xiyj) + h(xiyj) = t + s = q + 1 for edge-image coefficient k00 = q + 1, see Figure2.

5 3 9 13 11 13 7 3 2 4 15 11

5 9 3 11 5 9 3 11 12 8 14 6 10 12 6 4 10 14 12 1 5 6 2 6 6 14 11 12 2 10 12 2 7 5 15 10 4 7 13 14 0 4 2 16 12 10 3 14 8 8 4 14 8 8 3 9 16 13 4 13 15 16 1 7 15 16 1 7 2 1 16 10

1 15 16 7 11 15 1 0 9 5 6 9 8 0 13 (a) f (b) g (c) h Figure 2. (a) A set-ordered graceful labeling f of a tree T;(b) A set-ordered graceful labeling g of T holding f (v) + g(v) = p − 1 = 16;(c) A set-ordered graceful labeling h of T holding f (uv) + h(uv) = q + 1 = 17.

The proof of the Theorem1 is complete.

Theorem 2. If a tree T admits set-ordered graceful labeling, then T holds the following assertions: (1) T admits a matching of odd-graceful v-image-labelings and a matching of odd-graceful e-image-labelings. (2) T admits a matching of odd-elegant v-image-labelings and a matching of odd-elegant e-image-labelings. (3) T admits a matching of (k, d)-graceful v-image-labelings, and a matching of (k, d)-graceful e-image-labelings. (4) T admits a (k, 2)-arithmetic total labeling and a (k + 2, 2)-arithmetic total labeling are a matching of v-image-labelings; T admits a matching of (k, 2)-arithmetic total e-image-labelings. (5) T admits a matching of super edge-magic total v-image-labelings and a matching of super edge-magic total e-image-labelings.

Proof. (1) We define a matching of odd-graceful v-image-labelings f1 and g1 from the set-ordered graceful v-image-labelings f and g. Letting f1(xi) = 2 f (xi) = 2i − 2 with i ∈ [1, s], f1(yj) = 2 f (yj) − 1 = 2s + 2j − 3 with j ∈ [1, t], and f1(xiyj) = f1(yj) − f1(xi) = 2(s − i + j) − 1 for each edge xiyj ∈ E(T). Since f (V(T)) = [0, q] holds true, then the set of vertex labels of T under the labeling f1 is a subset of [0, 2q − 1], and

o f1(xiyj) = | f1(xi) − f1(yj)| = |2 f (xi) − (2 f (yj) − 1)| = 2 f (xiyj) − 1 ∈ [1, 2q − 1] .

For any two vertices u, v ∈ V(T), f1(u) 6= f1(v) since f (u) 6= f (v), we claim that f1 is an odd-graceful labeling. Then, we define another odd-graceful labeling g1 by setting g1(xi) = 2(s + t − 1) − 1 − f1(xi) = 2(p − i) − 1 with i ∈ [1, s], g1(yj) = 2(s + t − 1) − 1 − f1(yj) = 2(t − j) with j ∈ [1, t], and g1(xiyj) = g1(xi) − g1(yj) = 2(s − i + j) − 1 for each edge xiyj ∈ E(T). Immediately, the vertex label set g1(V(T)) ⊆ [0, 2q − 1], and the vertex label set is the union of an odd numbers set and an even numbers set, so

o g1(xiyj) = |g1(xi) − g1(yj)|, g1(E(T)) = [1, 2q − 1] ,

we get f1(v) + g1(v) = 2(s + t − 1) − 1 = 2p − 3 for each vertex v ∈ V(T) and claim that 0 f1 and g1 are a matching of odd-graceful v-image-labelings with k = 2p − 3. Mathematics 2021, 9, 1393 5 of 8

We also define another odd-graceful labeling h1 by setting h1(xi) = 2(s − 1) − f1(xi) = 2(s − i) ≤ 2(s − 1) with i ∈ [1, s], 2s − 1 ≤ h1(yj) = 2(p + s − 2) − f1(yj) = 2(p − j) − 1 with j ∈ [1, t], so h1(V(T)) ⊆ [0, 2q − 1], h1(xiyj) = h1(yj) − h1(xi) = 2t + 2i − 2j − 1 for each edge xiyj ∈ E(T). In addition, h1(xiyj) = |h1(xi) − h1(yj)| = 2q − f2(xiyj), because o o f1(xiyj) ∈ [1, 2q − 1] , so h1(E(T)) = [1, 2q − 1] , h1(u) 6= h1(v) for distinct u, v ∈ V(T). Therefore, h1 is also an odd-graceful labeling, there is f1(xiyj) + h1(xiyj) = 2q for each edge xiyj ∈ E(T), so f1 and h1 are a matching of set-ordered odd-graceful e-image-labelings with edge-image coefficient k00 = 2q. The assertion (1) has been proven. (2) Setting f2(xi) = 2 f (xi) = 2i − 2 with i ∈ [1, s], f2(yj) = (s + p − 1) − f (yj) − j + 2 = p − 2j + 2 with j ∈ [1, t], since f2(xi) ≤ 2(s − 1) and f2(yj) ≤ p, so f2(V(T)) ⊆ [0, 2q − 1]. We set f2(xiyj) = f2(xi) + f2(yj)(mod 2q) = p + 2i − 2j (mod 2q) for each edge xiyj ∈ E(T), we can find f2(yj) is odd, and f2(xi) is even, so f2(xiyj) is odd, thus o f2(E(T)) = [1, 2q − 1] , so f2 is an odd-elegant labeling. There is another odd-elegant labeling g2 by setting g2(xi) = p − f2(xi) = p − 2i + 2 with i ∈ [1, s], and g2(yj) = p − f2(yj) = 2j − 2 with j ∈ [1, t], we can see 0 ≤ g2(xi), g2(yj) ≤ p and g2(V(T)) ⊆ [0, 2q − 1]. In addition, g2(xiyj) = g2(xi) + g2(yj)(mod 2q) = p − 2i + 2j (mod 2q) for each edge xiyj ∈ E(T), because the parity of g2(xi) and g2(yj) is opposite, so g2(xiyj) is odd, then we o get g2(E(T)) = [1, 2q − 1] , we also get f2(v) + g2(v) = p for v ∈ V(T), which indicate 0 that f2 and g2 are a matching of odd-elegant v-image-labelings with k = p. An odd-elegant labeling h2 of T can be obtained from f2 in the following way: h2(xi) = 2(s − 1) − f2(xi) = 2(s − i) with i ∈ [1, s], h2(yj) = 2t − f2(yj) = 2t − p + 2j − 2 with j ∈ [1, t], thus, the vertex label set h2(V(G)) ⊆ [0, 2q − 1]; since h2(xiyj) = h2(xi) + h2(yj)(mod 2q) = p − 2i + 2j − 2 (mod 2q) for each edge xiyj ∈ E(T), we get the edge la- o bel set h2(E(T)) = [1, 2q − 1] . Additionally, it is not hard to compute f2(xiyj) + h2(xiyj) = 2p − 2 = 2q for xiyj ∈ E(T), thus f2 and h2 are a matching of odd-elegant e-image-labelings with k00 = 2q. The assertion (2) has been proven. (3) For integers k ≥ 0 and d ≥ 1, we define a labeling f3 as: f3(xi) = d f (xi) = d(i − 1) with i ∈ [1, s], f3(yj) = d[ f (yj) − 1] + k = d(s + j − 2) + k with j ∈ [1, t]. Since f3(xi) ≤ d(s − 1) and s + j − 2 ≤ s + t − 2 = q − 1, we can get f3(yj) ≤ k + d(q − 1), and the vertex label set under f3 is a subset of [0, k + (q − 1)d]. Moreover,

f3(xiyj) = | f3(xi) − f3(yj)| = f3(yj) − f3(xi) = k + d(s + j − i − 1) ∈ Sk,d,

which shows that f3 is a (k, d)-graceful labeling of T. There is another (k, d)-graceful labeling g3 defined as: g3(xi) = k + (q − 1)d − f3(xi) = k + d(q − i) ≤ k + (q − 1)d with i ∈ [1, s], g3(yj) = k + (q − 1)d − f3(yj) = d(q − s − j + 1) with j ∈ [1, t], and g3(xiyj) = |g3(xi) − g3(yj)| = g3(xi) − g3(yj) = k + d(s − i + j − 1) for every edge xiyj ∈ E(T). We obtain the vertex label set g3(V(T)) ⊆ [0, k + (q − 1)d], and the edge label set g3(E(T)) = Sk,d, moreover, f3(v) + g3(v) = k + (q − 1)d holds true for v ∈ V(T), so we claim that f3 0 and g3 are a matching of (k, d)-graceful v-image-labelings with k = k + (q − 1)d. Another (k, d)-graceful labeling h3 defined as: h3(xi) = d(s − 1) − f3(xi) = d(s − i) with i ∈ [1, s], h3(yj) = 2k + d(2s + t − 3) − f3(yj) = k + d(p − j − 1) with j ∈ [1, t], h3(xiyj) = h3(yj) − h3(xi) = k + d(t + i − j − 1) for every edge xiyj ∈ E(T), h3(V(T)) ≤ [0, k + (q − 1)d], so

h3(E(T)) = {h3(xiyj) = | f3(xi) − f3(yj)| : xiyj ∈ E(T)} = Sk,d.

00 We can get k = f3(xiyj) + h3(xiyj) = 2k + (q − 1)d, which shows that f3 and h3 are a matching of (k, d)-graceful e-image-labelings with edge-image coefficient k00 = 2k + (q − 1)d. We have shown the assertion (3). (4) We, by defining a new labeling f4, set a transformation: f4(xi) = 2 f (xi) = 2(i − 1) ≤ 2(s − 1) with i ∈ [1, s], f4(yj) = k + 2[ f (yj) − 2j + 2] = k + 2(t − j) with j ∈ [1, t], then k ≤ f4(yj) ≤ k + 2(t − 1), so f4(u) 6= f4(v) for distinct u, v ∈ V(T), and f4(V(T)) ⊆ [0, k + 2(q − 1)], on the other hand, f4(xiyj) = k + 2(t − j + i − 1) for each edge xiyj ∈ E(T), we calculate the set { f4(xiyj) = f4(xi) + f4(yi) : xiyj ∈ E(T)} = Sk,2, Mathematics 2021, 9, 1393 6 of 8

which means that the labeling f4 is a (k, 2)-arithmetic total labeling of T. Another labeling g4 differing from f4 is defined as: g4(xi) = k + 2(t − 1) − f4(xi) = k + 2t − 2i with i ∈ [1, s], g4(yj) = k + 2(t − 1) − f4(yj) = 2j − 2 with j ∈ [1, t], and g4(xiyj) = k + 2t − 2i + 2j − 2 for each edge xiyj ∈ E(T), furthermore, the vertex label set g4(V(T)) ⊆ [0, k + 2(q − 1)], and

{g4(xiyj) = g4(xi) + g4(yj) : xiyj ∈ E(T)} = {k + 2, k + 4, k + 6, . . . , k + 2q}.

So, g4 is a (k + 2, 2)-arithmetic total labeling, in addition, f4(v) + g4(v) = k + 2(t − 1) for each vertex v ∈ V(T), this means that f4 and g4 are a matching of v-image-labelings with k0 = k + 2(t − 1). We come to define a (k, 2)-arithmetic total labeling h4 of T as follows: h4(xi) = f (xi) + 2s − 3i + 1 = 2(s − i) with i ∈ [1, s], h4(yj) = f (yj) + j − s + 2 = 2j + 1 with j ∈ [1, t], h4(xiyj) = 2(s − i + j) + 1 for each edge xiyj ∈ E(T), the vertex label set h4(V(G)) ⊆ [0, k + 2(q − 1)], and moreover

{h4(xiyj) = h4(xi) + h4(yj) : xiyj ∈ E(T)} = Sk,2.

Then, we considering f4(xiyj) + h4(xiyj) = k + 2(t − j + i − 1) + 2(s − i + j) + 1 = k + 2q + 1, so labelings f4 and h4 are a matching of (k, 2)-arithmetic total e-image-labelings with edge-image coefficient k00 = k + 2q + 1. The assertion (4) holds true. (5) There is a labeling f5 defined in the following way: f5(xi) = f (xi) + 1 = i with i ∈ [1, s], and f5(yj) = f (yj) + t − 2j + 2 = p − j + 1 with j ∈ [1, t], and we set f5(xiyj) = p + s − i + j for each edge xiyj ∈ E(T). It is not difficult to compute f5(xi) + f5(yj) + f5(xiyj) = 2p + s + 1, since 1 ≤ f5(xi) ≤ s, s + 1 ≤ f5(yj) ≤ p and p + 1 ≤ f5(xiyj) ≤ 2p − 1 = p + q, we have f5(V(G)) ∪ f5(E(G)) = [1, p + q], so f5 is a super edge-magic total labeling with magic coefficient 2p + s + 1 by definition. We come to define an edge-magic total labeling g5 in the way: p − s + 1 ≤ g5(xi) = p + 1 − f5(xi) = p − i + 1 ≤ p with i ∈ [1, s], 1 ≤ g5(yj) = p + 1 − f5(yj) = j ≤ t with j ∈ [1, t], g5(xiyj) = 2p + q + 1 − f5(xiyj) = t + q + i − j + 1 for each edge xiyj ∈ E(T), notice that p + 1 ≤ f5(xiyj) ≤ p + q, so condition g5(V(T)) ∪ g5(E(T)) = [1, p + q] is true, and

g5(xi) + g5(yj) + g5(xiyj) = p + 1 − f5(xi) + p + 1 − f5(yj) + 2p + q + 1 − f5(xiyj) (2) = 2p − s + q + 2

to be a constant. Hence, g5 is a super edge-magic total labeling, moreover, f5(v) + g5(v) = p + 1 for each v ∈ V(T) holds true, immediately, both labelings f5 and g5 are a matching of super edge-magic total v-image-labelings with k0 = p + 1. Another set-ordered edge-magic total labeling h5 with magic coefficient C = s + 2p + 1 is defined as: h5(xi) = s + 1 − f5(xi) = s − i + 1 with i ∈ [1, s], h5(yj) = 2s + t + 1 − f5(yj) = s + j with j ∈ [1, t], h5(xiyj) = 2p + i − j − s for each edge xiyj ∈ E(T). We also 00 obtain h5(xi) + h5(xiyj) + h5(yj) = s + 2p + 1 = C. In addition, there is k = f5(xiyj) + h5(xiyj) = 3p = 3(q + 1) for each edge xiyj ∈ E(T), which means that f5 and h5 are a matching of super set-ordered edge-magic total e-image-labelings with edge-image coefficient k00 = 3(q + 1). This is the proof of the assertion (5).

For understanding the proof of Theorem2, see the several matching-type image- labelings of a tree shown in Figure3. If the two labelings f and l constituting the image- labelings are not the same type, we get a result as follows:

Corollary 1. If a tree T admits a set-ordered graceful labeling f , then it admits an edge-magic total labeling l, so that f and l are a matching of e-image-labelings with k00 = p + q + 1.

Proof. Let (X, Y) be the bipartition of vertices of a tree T, where X = {xi : i ∈ [1, s]} and Y = {yj : j ∈ [1, t]} holding s + t = |V(T)| = p. By the hypothesis of the theorem, T admits a graceful labeling f such that f (xi) = i − 1 with i ∈ [1, s], f (yj) = s + j − 1 with j ∈ [1, t] and f (xiyj) = f (yj) − f (xi) = s + j − i for each edge xiyj ∈ E(T), as well Mathematics 2021, 9, 1393 7 of 8

as f (V(T)) = [0, q] and f (E(T)) = [1, q], again, max f (X) = s − 1 < min f (Y) = s, so f is a set-ordered graceful labeling. Next, we define an edge-magic total labeling l: l(xi) = p − f (xi) = p − i + 1 with i ∈ [1, s], l(yj) = p − f (yt−j+1) = j with j ∈ [1, t], and l(xiyj) = q + t + i − j + 1 for every edge xiyj ∈ E(T), as well as l(V(T)) ∪ l(E(T)) = [1, p + q], and we have l(xi) + l(xiyj) + l(yj) = p + q + t + 2 = 2p + t + 1 = C is a positive constant, thus l is called an edge-magic total labeling. In addition, we can see f (xiyj) + l(xiyj) = s − i + j + q + t + i − j + 1 = p + q + 1 is a constant, so f and l are a matching of e-image-labelings with k00 = p + q + 1.

10 6 17 25 21 25 14 6 4 8 29 21 10 6 15 7

9 17 5 21 9 17 5 21 23 15 27 11 23 15 27 11 19 23 12 8 19 27 23 2 10 9 12 11 4 12 11 27 21 13 21 4 9 9 19 23 3 19 23 31 3 13 14 29 13 0 29 27 0 8 4 23 19 6 5 8 27 15 7 27 15 16 7 5 17 31 25 5 1717 25 15 25 7 25 29 31 1 13 29 31 1 13 3 1 31 19 7 3 1 31 19

2 29 31 14 21 29 2 0 17 10 12 17 15 0 25 2 3 1 14 11 g1 f1 h1 f2 7 11 2 10 4 8 3 11 15 9 27 39 33 39 21 9

11 19 7 23 9 17 5 21 15 27 9 33 15 27 9 33 8 5 5 9 2 10 30 36 18 12 30 4 13 13 11 18 6 18 18 42 21 25 5 19 23 3 36 6 30 36 6 17 14 30 48 12 12 9 13 6 42 0 12 6 36 29 17 0 9 27 15 1 7 42 24 12 42 24 24 27 25 24 39 31 1 3 15 29 31 1 13 39 45 48 3 21 45 48 3 21

15 14 16 3 6 12 15 17 0 7 3 45 48 21 33 45 3 0 27 15 g2 h2 f3 g3 6 12 45 33 10 6 17 9 9 13 2 10 4 8 5 13

36 24 42 18 25 17 29 13 13 21 9 25 11 19 7 23 36 3 15 11 12 4 8 7 15 7 11 2 10 42 33 15 23 4 15 13 21 15 45 15 11 31 23 27 7 21 25 5 39 19 11 14 9 30 21 9 7 0 8 12 11 15 6 9 27 48 7 1919 27 31 17 0 29 17 3 12 29 27 33 6 3 48 30 9 5 3 33 21 33 35 5 17 31 3 15

18 27 24 0 39 2 5 3 14 13 17 14 16 5 6 12 17 19 0 9 g h3 f4 4 h4 6 4 16 12 12 14 2 6 3 5 10 14

22 26 20 28 29 25 31 23 29 25 31 23 13 7 3 5 11 15 11 13 2 6 15 23 3 28 28 27 29 19 24 22 32 24 22 32 21 17 30 8 30 11 1 5 7 13 15 4 31 25 17 20 26 1 20 26 9 30 21 21 18 32 33 18 24 19 18 33 27 19 33 27

2 10 9 8 14 16 8 9 10 4 7 16 17 1 12

f5 g5 h5

Figure 3. The examples for illustrating the conversions between image-labelings fs, fs, hs with s = 1, 2, 3, 4, 5 in the proofs of Theorem2.

3. Conclusions Inspired by public and private keys in graphical passwords, we propose two new labelings in this paper, called vertex image-labelings and edge image-labelings respectively. We combine the new labelings with the known labelings to form compound labelings to find the relationships between the two compound labelings. Starting from the set- ordered graceful labeling f of the tree, we first prove that two different set-ordered graceful labelings match them to form a matching of vertex image-labeling and a matching of edge image-labelings. Then, according to labeling f , the odd-graceful image-labelings, odd-elegant image-labelings, (k, d)-graceful image-labelings, (k, d)-arithmetic total image- labelings, super edge-magic total image-labelings of the tree are derived in turn. The new Mathematics 2021, 9, 1393 8 of 8

image-labelings can connect two labelings of the same type. This paper draws a conclusion: the above-mentioned matching-type image-labelings of the tree can be transformed into each other, knowing one of the labelings of a tree, we can quickly get the other labelings of the tree.

Author Contributions: Create and conceptualize the idea, B.Y. and J.S.; writing—original draft preparation, J.S. and H.W.; writing—review and editing, J.S. and B.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the National Natural Science Foundation of China under grants No. 61363060, No. 61662066, No. 61902005, and China Postdoctoral Science Foundation Grants No. 2019T120020 and No. 2018M641087. Conflicts of Interest: The authors declare no conflict of interest.

References 1. Ringel, G. Problem 25 in theory of graphs and its applications. Proc. Symp. Smolenice 1963, 162. 2. Kumar, A.; Vats, A.K. Application of graph labeling in crystallography. Mater. Today Proc. 2020, 3.[CrossRef] 3. Prasanna, N.L.; Sravanthi, K.; Sudhakar, N. Applications of graph labeling in communication networks. Orient. J. Comput. Sci. Technol. 2014, 7, 139–145. 4. Tian, Y.; Li, L.; Peng, H.; Yang, Y. Achieving Flatness: Graph labeling can generate graphical honeywords. Comput. Secur. 2021, 5, 102212. [CrossRef] 5. Van, D.; Leese, R.A.; Shepherd, M.A. Graph labeling and radio channel assignment. J. Graph. Theory 2015, 29, 263–283. 6. Gallian, J.A. A dynamic survey of graph labeling. Electron. J. Comb. 2014, 17, 60–62. 7. Rosa, A. On Certain Valuations of the Vertices of a Graph. In Theory of Graphs: International Symposium, Rome, Italy, July 1966; Gordon and Breach: New York, NY, USA, 1967; pp. 349–355. 8. Gnanajothi, R.B. Topics in Graph Theory. Ph.D. Thesis, Madurai Kamaraj University, Tamil Nadu, India, 1991. 9. Zhou, X.Q.; Yao, B.; Chen, X.E.; Tao, H.X. A proof to the odd-gracefulness of all lobsters. Ars Comb. 2012, 103, 13–18. 10. Chang, G.J.; Hsu, D.F.; Rogers, D.G. Additive variations on a graceful theme: Some results on harmonious and other related graphs. Congr. Numer. 1981, 32, 181–197. 11. Ringel, G.; Llado, A.S. Another tree conjecture. Bull. ICA 1996, 18, 83–85. 12. Enomoto, H.; Llado, A.S.; Nakamigawa, T.; Ringel, G. Super Edge-magic Graphs. Sut J. Math. 1998, 2, 105–109. 13. Acharya, B.D.; Hegde, S.M.J. Graph theory. Arith. Graphs 1990, 14, 275–299. 14. Dinnen, M.J.; Ke, N.; Khosravani, M. Arithmetic progression graphs. Univ. J. Appl. Math. 2014, 2, 290–297. [CrossRef] 15. Gao, Y.P.; Shan, S.L. Antimagic orientation of lobsters. Discret. Apllied Math. 2020, 287, 21–26. [CrossRef] 16. Sethuraman, G.; Murugan, V. Generating graceful unicyclic graphs from a given forest. AKCE Int. J. Graphs Comb. 2020, 17, 592–605. [CrossRef] 17. Sabirov, D.; Tukhbatullina, A.; Shepelevich, I. Information entropy of regular dendrimer aggregates and irregular intermediate structures. Liquids 2021, 1, 25–35. [CrossRef] 18. Wang, H.Y.; Xu, J.; Ma, M.Y.; Zhang, H.Y. A new type of graphical passwords based on odd-elegant labelled graphs. Secur. Commun. Netw. 2018, 2018, 1–11. [CrossRef] 19. Bondy, J.A.; Murty, U.S.R. Graph Theory; Springer: Berlin/Heidelberg, Germany, 2008. 20. Golomb, S.W. How to Number a Graph. In Graph Theory and Computing; Read, R.C., Ed.; Academic Press: Cambridge, MA, USA, 1972; pp. 23–37.