Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is, if we have $Sum(u) \neq Sum(v)$ whenever $u$ and $v$ are adjacent. The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\lbrace 1, 2, ...,k\rbrace$ is the lucky number of $G$, denoted by $η(G)$. We will prove, for every graph $G$ other than $ K_{2} $, $\frac{w}{n-w+1}\leqη(G) \leq Δ^{2} $ and we present an algorithm for lucky labeling of $ G $.
If $\ell: V(G)\rightarrow {\mathbb N}$ is a vertex labeling of a graph $G = (V(G), E(G))$, then the $d$-lucky sum of a vertex $u\in V(G)$ is $d_\ell(u) = d_G(u) + \sum_{v\in N(u)}\ell(v)$. The labeling $\ell$ is a $d$-lucky labeling if $d_\ell(u)\neq d_\ell(v)$ for every $uv\in E(G)$. The $d$-lucky number $η_{dl}(G)$ of $G$ is the least positive integer $k$ such that $G$ has a $d$-lucky labeling $V(G)\rightarrow [k]$. A general lower bound on the $d$-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinit…
A coprime labeling of a graph of order $n$ is an assignment of distinct positive integer labels in which adjacent vertices have relatively prime labels. Restricting labels to only the set $1$ to $n$ results in a prime labeling. In this paper, we consider families of graphs in which a prime labeling cannot exist with the goal being to minimize the largest value of the labeling set, resulting in a minimum coprime labeling. In particular, prism graphs, generalized Petersen graphs with $k=2$, and stacked prism graphs are investigated for minimum coprime labelings.
Given a graph $G$, a total labeling on $G$ is called edge-antimagic total (respectively, vertex-antimagic total) if all edge-weights (respectively, vertex-weights) are pairwise distinct. If a labeling on $G$ is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graph…
An odd prime labeling is a variation of a prime labeling in which the vertices of a graph of order~$n$ are labeled with the distinct odd integers $1$ to $2n-1$ so that the labels of adjacent vertices are relatively prime. This paper investigates many different classes of graphs including disjoint unions of cycles, stacked prisms, and particular types of caterpillars, by using various methods to construct odd prime labelings. We also demonstrate progress toward proving a conjecture that all prime graphs have an odd prime labeling.
Since Sedlá$\breve{\mbox{c}}$ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected bipartite graph. We make a very useful observation that there are only four possible values of $b$ for which a connected bipartite graph has a $b$-edge consecutive magic labeling. On the basis of this fundamental result, we deduce various interesting results on consecutive edge magic labelings of bipartite graphs, especially caterpillars and lobsters, which exten…
This paper deals with the problem of finding totally antimagic total labelings of complete bipartite graphs. We prove that complete bipartite graphs are totally antimagic total graphs. We also show that the join of complete bipartite graphs with one vertex is a totally antimagic total graph
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.
In 2006 Qian [J. Qian, Degree complete graphs; Discrete Mathematics 306 (2006), 533--537] introduced the concept of degree complete graphs for labeled graphs. He also gave a characterization of these graphs in terms of two forbidden subgraphs. Furthermore, he mentioned that the property of being degree complete depends on the labeling of the graph. Related to this he stated the problem to find a characterization of those (unlabeled) graphs for which every labeled version is not degree complete. We say that a (unlabeled) graph has a degree complete labeling, if there is a labeled version of the…
We prove the existence of infinite classes of cyclic G-decompositions of the complete multipartite graph, G being a caterpillar, a hairy cycle or a cycle. All the results are obtained by the construction of d-divisible $α$-labelings of G, introduced in [A. Pasotti, On d-graceful labelings, to appear on Ars Combin.] as a generalization of classical $α$-labelings, whose existence implies that one of graph decompositions.
A prime labeling of a graph of order $n$ is a labeling of the vertices with the integers $1$ to~$n$ in which adjacent vertices have relatively prime labels. A coprime labeling maintains the same criterion on adjacent vertices using any set of distinct positive integers. In this paper, we consider several families of graphs or products of graphs that have been shown to not have prime labelings and answer the natural question of how to label the vertices while minimizing the largest value in its set of labels.
We extend a result of Griggs and Yeh about the maximum possible value of the L(2,1)-labeling number of a graph in terms of its maximum degree to oriented graphs. We consider the problem both in the usual definition of the oriented L(2,1)-labeling number and in some variants we introduce.
Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than $1$, the greatest common divisor of the set of labels in the neighborhood of $v$ is $1$. In this paper, we introduce techniques for finding neighborhood-prime labelings based on the Hamiltonicity of the graph, by using conditions on possible degrees of vertices, and by examining a neighborhood graph. In particular, classes of graphs shown to be neighborhood-prime include all generalized Petersen graphs, grid grap…
In this article we introduce the notion of weak harmonic labeling of a graph, a generalization of the concept of harmonic labeling defined recently by Benjamini et al. that allows extension to finite graphs and graphs with leaves. We present various families of examples and provide several constructions that extend a given weak harmonic labeling to larger graphs. In particular, we use finite weak models to produce new examples of (strong) harmonic labelings. As a main result, we provide a characterization of weakly labeled graphs in terms of harmonic subsets of the integers and use it to compu…
In this paper we formulate cycle-supermagic labelings for the disjoint union of isomorphic copies of different families of graphs. We also prove that disjoint union of non isomorphic copies of fans and ladders are cycle-supermagic.
A neighborhood-prime labeling of a graph is a variation of a prime labeling in which the vertices are assigned labels from $1$ to $|V(G)|$ such that the $\gcd$ of the labels in the neighborhood of each non-degree $1$ vertex is equal to $1$. In this paper, we examine neighborhood-prime labelings of several classes of graphs such as polygonal snakes and books, with a focus on trees including caterpillars, spiders, and firecrackers.
A coprime labeling of a simple graph of order $n$ is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first $n$ positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss coprime labelings for complete bipartite graphs.
The Antimagic Graph Conjecture asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, \dots, |E|$ is used exactly once and the sums of the labels on all edges incident to a given vertex are distinct. On the other extreme, an edge labeling is magic if the sums of the labels on all edges incident to each vertex are the same. In this paper we approach antimagic labelings by introducing partially magic labelings, where "magic occurs'' just in a subset of $V$. We generalize Stanley's theorem about the magic graph labeling counting function t…
A simple $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$ such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, all of which are special cases of Seoud and Youssef's conjecture that all unicyclic graphs have a prime labeling.
In this paper, we obtained that every 5-regular graph admits a zero-sum 3-magic labeling, which give an affirmative answer to a conjecture proposed by Saieed Akbari, Farhad Rahmati and Sanaz Zare in $Electron.$ $J.$ $Combin.$.
A coprime labeling of a graph $G$ is a labeling of the vertices of $G$ with distinct integers from $1$ to $k$ such that adjacent vertices have coprime labels. The minimum coprime number of $G$ is the least $k$ for which such a labeling exists. In this paper, we determine the minimum coprime number for several well-studied classes of graphs, including the coronas of complete graphs with empty graphs and the joins of two paths. In particular, we resolve a conjecture of Seoud, El Sonbaty, and Mahran and two conjectures of Asplund and Fox. We also provide an asymptotic for the minimum coprime numb…
For $k\in\mathbb{Z}^+$ and $G$ a simple connected graph, a $k$-radio labeling $f:V_G\to\Z^+$ of $G$ requires all pairs of distinct vertices $u$ and $v$ to satisfy $|f(u)-f(v)|\geq k+1-d(u,v)$. When $k=1$, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different "colors". We consider $k$-radio labelings of $G$ when $k=\diam(G)$. In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibiliti…
For a graph $G$, a $k$-radio labeling of $G$ is the assignment of positive integers to the vertices of $G$ such that the closer two vertices are on the graph, the greater the difference in labels is required to be. Specifically, $\vert f(u)-f(v)\vert\geq k + 1 - d(u,v)$ where $f(u)$ is the label on a vertex $u$ in $G$. Here, we consider the case when $G$ is the Cartesian products of complete graphs. Specifically we wish to find optimal labelings that use consecutive integers and determine when this is possible. We build off of a paper by Amanda Niedzialomski and construct a framework for disco…
It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-$k$-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference. Let $k\geq1$ be an integer and $ψ$ is a labeling function (anti-$k$-labeling) from …
We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinator…
A labeling of a graph is a bijection from $E(G)$ to the set $\{1, 2,..., |E(G)|\}$. A labeling is \textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem.
The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called $k$-shifted-antimagic labelings which allow the consecutive numbers starting from $k+1$, instead of starting from 1, where $k$ can be any integer. This paper establishes connections among various concepts proposed in …
A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.
Technical University of Ostrava（俄斯特拉发工业大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Primorska, FAMNIT（沿海大学FAMNIT学院）；University of Ljubljana, Faculty of Education（卢布尔雅那大学教育学院）；University of Primorska, Institute Andrej Marušič（沿海大学安德烈亚·马鲁西奇研究所）
A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.
A good edge-labeling of a graph [Araújo, Cohen, Giroire, Havet, Discrete Appl. Math., forthcoming] is an assignment of numbers to the edges such that for no pair of vertices, there exist two non-decreasing paths. In this paper, we study edge-labeling on graphs with girth at least 5. In particular we verify, under this additional hypothesis, a conjecture by Araújo et al. This conjecture states that if the average degree of G is less than 3 and G is minimal without an edge-labeling, then G \in {C_3,K_{2,3}}. (For the case when the girth is 4, we give a counterexample.)
An antimagic labeling for a graph $G$ with $m$ edges is a bijection $f: E(G) \to \{1, 2, \dots, m\}$ so that $ϕ_f(u) \neq ϕ_f(v)$ holds for any pair of distinct vertices $u, v \in V(G)$, where $ϕ_f(x) = \sum_{x \in e} f(e)$. A strongly antimagic labeling is an antimagic labeling with an additional condition: For any $u, v \in V(G)$, if $°(u) > °(v)$, then $ϕ_f(u) > ϕ_f(v)$. A graph $G$ is strongly antimagic if it admits a strongly antimagic labeling. We present inductive properties of strongly antimagic labelings of graphs. This approach leads to simplified proofs that spiders and double spide…
We prove that if $G$ is a $2r$-regular edge graceful $(p,q)$ graph with $(r,kp)=1$ then $kG$ is edge graceful for odd $k$. We also prove that for certain specific classes of $2r$-regular edge graceful graphs it is possible to drop the requirement that $(r,kp)=1$
In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the iterated Mycielski of graphs in general. For a graph $G$ and all $t\geq 1$, we give sharp bounds for $λ(M^t(G))$ the $L(2, 1)$-labeling number of the $t$-th iterated Mycielski in terms of the number of iterations $t$, the order $n$, the maximum degree $\bigtriangleup$, and $λ(G)$ the $L(2, 1)$-labeling number of $G$. For $t=1$, we present necessary and sufficient conditions between the $4$-star matching number of the complement graph and $λ(M(G))$ the $L(2, 1)$-labeling number of the Mycielski of a graph, with some applica…
Magic labelings of graphs are studied in great detail by Stanley and Stewart. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
Let $G=(V,E)$ be a graph of order $n$. A closed distance magic labeling of $G$ is a bijection $\ell \colon V(G)\rightarrow \{1,\ldots ,n\}$ for which there exists a positive integer $k$ such that $\sum_{x\in N[v]}\ell (x)=k$ for all $v\in V $, where $N[v]$ is the closed neighborhood of $v$. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic labelings and the spectra of graphs. These results are then applied to the strong product of graphs with complete graph or cycle and to the circulant graphs. We end…
Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j} (n-i+1)\ge j{{n+1}\choose{2}}/k$ for all $j=1,...,k$. The conjecture is proved for $k=4$, extending earlier results for $k=2,3$.
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $f$ be a 0-1 labeling of $E(G)$ so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling $f$ \emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{partial vertex labeling} if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. Vertices that are incident to an equal number of edges of both labels we call \emph{unlabeled}. Call a procedure on a labeled graph…
A \textit{subtractive arc-magic labeling} (SAML) of a directed graph $G=(V,A)$ is a bijection $λ:V\cup A \to \{1,2,\ldots,|V|+|A|\}$ with the property that for every $xy\in A$ we have $λ(xy)+λ(y)-λ(x)$ equals to an integer constant. If $λ(xy)+λ(y)-λ(x)$ are distinct for every $xy\in A$, then $λ$ is a \textit{subtractive arc-antimagic labeling} (SAAL). A \textit{subtractive vertex-magic labeling} (SVML) of $G$ is such bijection with the property that for every $x\in V$ we have $λ(x)+\sum_{y\in V, yx \in A} λ(yx)-\sum_{y\in V, xy\in A} λ(xy)$ equals to an integer constant. If $λ(x)+\sum_{y\in V,…
Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$ is the neighborhood of $v$. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.
A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose as…
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \ne g^+(v)$, where $g^+(u) = \sum_{e\in E(u)} g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_f(u)\ne w_f(v)$…
A mapping $l : E(G) \rightarrow A$, where $A$ is an abelian group which written additively, is called a labeling of the graph $G$. For every positive integer $h \geqslant 2$, a graph $G$ is said to be zero-sum $h$-magic if there is an edge labeling $l$ from $E(G)$ into $\mathbb{Z}_{h} \backslash \{0\}$ such that $s(v) = \sum_{uv\in E(G)}l(uv) = 0$ for every vertex $v \in V(G)$. In 2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare conjectured that every 5-regular graph admits a zero-sum $3$-magic labeling. In this paper, we obtained that every 5-regular graph with every edge contains in a tria…
Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $μ$, we define the polynomial $$P_μ(A)=\sum_{σ\in S_n} a_{1σ(1)}\cdots a_{nσ(n)}\,μ^{\ell(σ)}\; ,$$ as the $μ$-permanent of $A$, where $\ell(σ)$ is the number of inversions of the permutation $σ$ in the symmetric group $S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $μ$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.
