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Matching-Type Image-Labelings of Trees 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 f0, 1, 2, ··· , qg 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 jVj 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 2 V1 and v2 2 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 jXj. We will use an integer set Sk,d = fk, k + d, k + 2d, ... , k + (q − 1)dg for integers k ≥ 0 and d ≥ 1, and use [x, y] to stand for an integer set fx, x + 1, ... , yg with two integers x, y subject to 0 ≤ x < y, as well as use [s, t]o to indicate an integer set fs, s + 2, ... , tg 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 f (u) : u 2 Sg by f (S), and we restate several well-known labelings as follows: Definition 1 ([20]). If the mapping q holds q(V(G)) ⊆ [0, q] with min q(V(G)) = 0, q(u) 6= q(v) for distinct u, v 2 V(G), and q(E(G)) = fq(uv) = jq(u) − q(v)j : uv 2 E(G)g = [1, q], then we call q a graceful labeling of G. If the (p, q)-graph G is a bipartite graph with vertex partition (X, Y), and a graceful labeling q holds maxfq(u) : u 2 Xg < minfq(v) : v 2 Yg (abbreviated as max q(X) < min q(Y)), we call q a set-ordered graceful labeling. Definition 2 ([6]). A (p, q)-graph G admits a mapping a holding a(V(G)) = f f (u) : u 2 V(G)g ⊆ [0, 2q − 1] and min a(V(G)) = 0, a(u) 6= a(v) for distinct vertices u, v 2 V(G), a(E(G)) = fa(uv) = ja(u) − a(v)j : uv 2 E(G)g = [1, 2q − 1]o, then a is called an odd- graceful labeling of graph G. Definition 3 ([6]). If a (p, q)-graph G has a function b holds b(V(G)) ⊆ [0, 2q − 1], as well as b(u) 6= b(v) for distinct vertices u, v 2 V(G), and b(E(G)) = fb(uv) = b(u) + b(v)(mod 2q) : uv 2 E(G)g = [1, 2q − 1]o, we call b an odd-elegant labeling of graph G. Definition 4 ([13]). If there is a labeling d of a (p, q)-graph G holding d(V(G)) ⊆ [0, k + (q − 1)d] such that d(u) 6= d(v) for distinct u, v 2 V(G), d(E(G)) = fd(uv) = jd(u) − d(v)j : uv 2 E(G)g = fk, k + d, k + 2d, ... , k + (q − 1)dg, then we call d 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 g of a (p, q)-graph G is said to be (k, d)-arithmetic total labeling if g(V(G)) ⊆ [0, k + (q − 1)d], g(u) 6= g(v) for distinct vertices u, v 2 V(G), and fg(u) + g(v) : uv 2 E(G)g = fk, k + d, k + 2d,..., k + (q − 1)dg holds. Mathematics 2021, 9, 1393 3 of 8 Definition 6 ([6]). If the mapping s holds s(V(G)) [ s(E(G)) = [1, p + q], and s(u) + s(v) + s(uv) = C for each edge uv 2 E(G), where C is a magic constant, then we call s an edge-magic total labeling of G; moreover, s is called a super edge-magic total labeling if s(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 2 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 2 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).
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