Graph labeling games
Zsolt Tuza
本文提出研究两人图标记博弈的多种新变体，并指出该领域目前几乎空白，为后续研究开辟方向。
我们提出研究两人图标记博弈的许多新变体。迄今为止，在这个广阔的开放领域中几乎没有任何研究。
We propose the study of many new variants of two-person graph labeling games. Hardly anything has been done in this wide open field so far.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Some Winnability Results for the Neighborhood and Group Labeling Lights Out Games
本文研究路径图、环图和完全二部图上的群标记 Lights Out 游戏，给出对任意初始标记均可获胜的充分必要条件，并为完全二部图上的邻域 Lights Out 游戏提供了新证明。
We study the value of unique games as a graph-theoretic parameter. This is obtained by labeling edges with permutations. We describe the classical value of a game as well as give a necessary and sufficient condition for the existence of an optimal assignment based on a generalisation of permutation graphs and graph bundles. In considering some special cases, we relate XOR games to EDGE BIPARTIZATION, and define an edge-labeling with permutations from Latin squares.
Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set $\{0,1,2,\dots, \ell-1\}$ for $\ell \in \mathbb{N}$. The game is played by toggling vertices: when a vertex is toggled, that vertex and each of its neighbors has its label increased by $1$ (modulo $\ell$). The game is won when every vertex has label 0. For any $n\in\mathbb{N}$ it is clear that one cannot win the game on $K_n$ unless the initial labeling assigns all vertices the same label. Given that the $K_n$ has the maximum number of edges of any simple graph on $n$ vertices…
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.
On consecutive edge magic total labeling of connected bipartite graphs
自从Sedlá $\breve{\mbox{c}}$ ek在1963年引入图的幻标号的概念以来，人们定义并研究了图的各种幻标号。本文研究了连通二部图的连续边幻标号。我们做了一个非常有用的观察，有只有四个可能的值的$b$，一个连接的二部图有一个$b$ -边连续魔术标签。在这个基本结果的基础上，我们推出了关于二部图，特别是毛虫图和龙虾图的连续边幻标号的各种有趣结果，推广了Sugen和Miller的结果.
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
Graph Labeling with Distance Conditions and the Delta Squared Conjecture
利用单图的阶和最大度给出了它的L（2，1）-标号数的界.我们还描述了一个无限类的图，其元素具有最高的L（2，1）-标号数方面的任何已知的无限类的图的最大程度。
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…
Decompositions of complete multipartite graphs via generalized graceful labelings
本文证明了完全多部图的无限类循环G-分解的存在性，G是毛毛虫、毛圈或圈。所有的结果都是通过构造G的d-可除$α$ -标号得到的。帕索蒂，关于d-优美的标签，出现在Ars Combin上。作为经典$α$ -标号的推广，它的存在性意味着图的一种分解.
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.
A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in whi…
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…
On cycle-supermagic labelings of the disconnected graphs
在本文中，我们制定了圈supermagic标签的不交并同构副本的不同家庭的图。我们还证明了扇和梯子的非同构副本的不交并是圈超幻的。
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.
Partially magic labelings and the Antimagic Graph Conjecture
反魔术图猜想断言除$K_2$外的所有连通图$G = (V, E)$都允许边标号，使得每个标号$1, 2, \dots, |E|$只使用一次，并且与给定顶点相关的所有边上的标号之和是不同的.在另一个极端，一个边的标签是魔术，如果在所有的边缘事件的标签的总和，每个顶点是相同的。在本文中，我们通过引入部分魔术标号来处理反魔术标号，其中“魔术”只发生在$V$的一个子集中。将Stanley关于幻图标号计数函数的定理推广到部分幻图标号的计数函数，并证明了它是周期至多为$2$的拟多项式.这使我们能够引入弱反魔术标号（允许重复），我们证明了每个二部图满足一个削弱版本的反魔术图猜想。
Prime Vertex Labelings Of Families Of Unicyclic Graphs
一个简单的$n$ -顶点图有一个素顶点标号，如果顶点可以用整数$1, 2, 3,\ldots, n$内射地标号，使得邻顶点有互素标号.我们将提出以前未知的素顶点标号为新的家庭的图，所有这些都是特殊情况下的Seoud和Youssef的猜想，所有的单圈图有一个素标签。
A Conjecture on Zero-sum 3-magic Labeling of 5-regular Graphs
本文证明了每个5-正则图都有一个零和3-幻标号，从而肯定地回答了Saieed Akbari，Farhad Rahmati和Sanaz Zare在$Electron.$ $J.$ $Combin.$中提出的一个猜想.
On the number of $2$-dice games with prime power dice-size
该研究针对素数幂尺寸的Sicherman骰子，利用生成多项式技术给出了满足和分布条件的二维骰子游戏数量的上下界，并证明了对应的非闭合组合公式。
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…
On Link-irregular labelings of Graphs
本文引入图的链不规则标号概念并定义标号数 $η(G)$，给出了该标号存在的充要条件，证明特定图族不容许该标号而完全图和轮图容许，并确定了相关标号数的精确或渐近值。
Random Walk Labelings of Perfect Trees and Other Graphs
本文延续近期研究，计算了完美树、梳图、双梳图、环面$C_2\ imes C_n$及由三条路图连接形成两个环的图的随机游走标号数量。
On Perrin Cordial Labeling: A New Approach in Graph Labeling Theory
本文提出了受Perrin数列启发的Perrin cordial标号概念，通过单射顶点标号及模2诱导边标号定义，并探讨了其在不同图族中的存在性，为数论与图结构建立了新联系。
Distance Magic Labeling of Generalised Mycielskian Graphs
本文针对若干图族，考察其广义 Mycielskian 图上的距离幻标号存在性与构造问题。
Total Prime Labelings of Various Graphs
本文提出了一种总素数标记方法，用于各种图类，证明了蛇形图、书形图、棱柱图等是总素数图。
Cartesian powers of graphs and consecutive radio labelings
对于简单连通图$k\in\mathbb{Z}^+$和$G$，$G$的$k$ -无线电标号$f:V_G\to\Z^+$要求所有不同顶点对$u$和$v$满足$|f(u)-f(v)|\geq k+1-d(u,v)$ .当$k=1$时，这个要求产生了称为顶点着色的熟悉的标记，对于该标记，图的每个顶点被标记为使得邻顶点具有不同的“颜色”。我们认为$k$-$G$的放射性标记时，$k=\diam(G)$。在这种情况下，没有两个顶点可以有相同的标签，所以具有连续整数的无线电标签的图是可能性频谱上的一个极端。这样的大直径图形的例子是特别罕见和可取的。我们构造任意高直径的例子，并进一步探索我们用来做这件事的工具--图的笛卡尔积--及其对无线电标签的影响。
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…
Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem
图的标号是从$E(G)$到集合$\{1, 2,..., |E(G)|\}$的双射。如果对于任何不同的顶点$u$和$v$，关联到$u$的边上的标签之和不同于关联到$v$的边上的标签之和，则标记为\textit{antimagic}。我们说一个图是反魔术的，如果它有一个反魔术标签。1990年，Ringel证明了除$K_2$之外的所有连通图都是反魔图。本文证明了正则二部图（度至少为2）是反幻图。我们的技术很大程度上依赖于婚姻定理。
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 …
Further Results on Signed Product Cordial Labeling
本文针对牛图分裂图、星图分裂图、路图平方图、Corona图及路径连接双Helm图等图类，研究了符号积亲切标号的存在性，扩展了该标号理论的适用范围。
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.
Further Results on Random Walk Labelings
本文进一步研究图的随机游走标号，计算了轮图、扇图、杠铃图、棒棒糖图、蝌蚪图、友谊图和蛇图等多个自然图族的随机游走标号数量，并证明了计算过程中出现的若干组合恒等式。
Self-reverse labelings of distance magic graphs
研究自反距离魔图的标签特性，提出新构造方法并确定阶数至30的连通四度图的自反距离魔标签情况。
Radio Labelings of Distance Graphs
一个连通图$G$的无线电$k$ -标记是一个非负整数的分配$c$到$G$的顶点，使得对于任何两个顶点$x$和$y$，$$|c(x) - c(y)| \geq k+1 - d(x,y),$$，$x\ne y$，其中$d(x,y)$是$G$中$x$和$y$之间的距离。本文研究了距离图的射电标号，即，以整数集$\Z$为顶点集的图，其中两个不同的顶点$i, j \in \Z$相邻当且仅当$|i - j| \in D$。
Good edge-labelings and graphs with girth at least five
图的好的边标记[Araújo，Cohen，Giroire，Havet，离散应用数学，[即将出版]是一个数字分配的边缘，使没有一对顶点，存在两个非递减路径。本文研究围长至少为5的图的边标号问题。特别是，我们验证，在这个额外的假设下，Araújo等人的猜想。这个猜想指出，如果G的平均度小于3和G是最小的没有边标签，那么G \in {C_3，K_{2,3} }。(For当周长为4时，我们给出一个反例。）
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…
On edge graceful labelings of disjoint unions of $2r$-regular edge graceful graphs
证明了如果$G$是一个$2r$ -正则边优美$(p,q)$图，且有$(r,kp)=1$，则$kG$对奇数$k$是边优美的.我们还证明了对于某些特定类的$2r$ -正则边优美图，可以放弃$(r,kp)=1$
The Lights Out Game on Directed Graphs
研究了有向图上的灯光熄灭游戏，证明了所有无环图在任意k下均可赢，并分析了竞赛图的可赢性条件。
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…
Cordial Labeling of Goldberg Snark and its related Graphs
本文针对图论中连通无桥且无法三色边着色的 Snark 图的修改结构 Goldberg Snark 图，研究其亲切标号问题，发现该图及其部分特殊形式均满足亲切标号。
On the Lucky labeling of Graphs
设图$G$的顶点被任意正整数标号，用$Sum(v)$表示顶点$v$的所有邻点标号之和.如果函数$Sum$是$G$的适当着色，也就是说，如果只要$u$和$v$相邻，我们就有$Sum(u) \neq Sum(v)$，则标记是幸运的。从集合$\lbrace 1, 2, ...,k\rbrace$中，图$G$具有幸运标记的最小整数$k$是$G$的幸运数，用$η(G)$表示。我们将证明，对于每一个图$G$以外的$ K_{2} $，$\frac{w}{n-w+1}\leqη(G) \leq Δ^{2} $，我们提出了一个算法的幸运标记$ G $。
Magic graphs and the faces of the Birkhoff polytope
图的幻标号由Stanley和Stewart详细研究。在本文中，我们使用多面体锥的Hilbert基和多面体的Ehrhart拟多项式构造并列举了图的魔标号。定义了图和有向图的幻标号的多面体。我们给出了一个描述的面的Birkhoff多面体的多面体的魔术标号的有向图。
Spectra of Graphs and Closed Distance Magic Labelings
设$G=(V,E)$是阶为$n$的图. $G$的闭距离幻标号是一个双射$\ell \colon V(G)\rightarrow \{1,\ldots ,n\}$，对于它存在一个正整数$k$，使得对于所有$v\in V $有$\sum_{x\in N[v]}\ell (x)=k$，其中$N[v]$是$v$的闭邻域。我们在代数的背景下考虑闭距离幻图。特别地，我们分析了闭距离魔标号与图的谱之间的关系。这些结果被应用到具有完全图或圈的图的强积和循环图。我们结束了一些理论问题，其解决方案的结果在另一个家庭的封闭距离魔术图有点相关的强产品。
Distance magic labeling in complete 4-partite graphs
设$G$是一个完全$k$ -部简单无向图，其部分大小为$p_1\le p_2...\le p_k$ .设$P_j=\sum_{i=1}^jp_i$为$j=1,...,k$。证明了$G$有距离魔标号当且仅当对所有$j=1,...,k$都有$\sum_{i=1}^{P_j} (n-i+1)\ge j{{n+1}\choose{2}}/k$.对$k=4$证明了该猜想，推广了以前对$k=2,3$的结果.
On the number of unlabeled vertices in edge-friendly labelings of graphs
设$G$是一个具有顶点集$V(G)$和边集$E(G)$的图，$f$是$E(G)$的0-1标号，使得标号为1和0的边的数目的绝对差不超过1。将这种标记称为$f$ \emph{edge-friendly}。如果与标记为1的边多于标记为0的边关联的顶点被标记为1，并且与标记为0的边多于标记为1的边关联的顶点被标记为0，则我们说边友好的标记会引发\emph{partial vertex labeling}。与两个标签的相等数量的边关联的顶点我们称为\emph{unlabeled}。如果一个过程是由成对的标签交换组成的，则称它为\emph{label switching algorithm}。给定一个边缘友好的标签$K_n$，我们展示了一个标签切换算法产生一个边缘友好的重新标记的$K_n$，这样所有的顶点被标记。我们称这种标记为\textit{opinionated}。
Prime Vertex Labelings of Several Families of Graphs
本文针对循环悬挂星、循环链、棱柱和广义书等新图族，给出了此前未知的素数顶点标号，即用1到n的整数单射标号顶点，使相邻顶点标号互素。
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 $λ(x)+\sum_{y\in V,…
On Geodesic Leech Labeling of Some Graph Classes
该文研究图的测地Leech标号问题，证明n≥5的圈为非测地Leech图、正则完全二部图中至多3个是测地Leech图，指出度序列无法刻画该类图，还求出轮图的测地路径数并给出W₅、W₆的对应标号。
Distance magic labeling and two products of graphs
设$G=(V,E)$是阶为$n$的图. $G$的距离魔标记是双射$\ell \colon V\rightarrow {1,...,n}$，对于该双射存在正整数$k$，使得对于所有$v\in V $都有$\sum_{x\in N(v)}\ell (x)=k$，其中$N(v)$是$v$的邻域。我们引入了距离幻图的一个自然子类。对于这个类，我们证明了它是封闭的直积与正则图和封闭的第二个因素的字典产品与正则图。此外，我们还刻画了两个圈的直积之间的距离幻图。
Antimagic labelings of a complete graph
本文利用Bhavale的边标号，证明了完全图K_n（n≥3）是超反魔法且全图全反魔法，并存在反魔法定向。
Graph Labelings Obtainable by Random Walks
本文开创了图的随机游走标号研究，通过随机游走按遇到未标号顶点的顺序递增标号，并利用二项式系数倒数和的新恒等式证明了 Bala 的一个猜想。
A Study on Topological Integer Additive Set-Labeling of Graphs
图$G$的集合标号是一个内射函数$f:V(G)\to \mathcal{P}(X)$，其中$X$是一个有限集合，$G$的集合索引子是一个集合标号，使得由$f^{\oplus}(uv) = f(u){\oplus}f(v)$对每个$uv{\in} E(G)$定义的导出函数$f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$也是内射的.设$G$是一个图，$X$是一个非空集.如果$f(V(G))$是$X$的一个拓扑，则集合索引器$f:V(G)\to \mathcal{P}(X)$称为$G$的一个拓扑集合标记。整数加性集合标号是一个单射函数$f:V(G)\to \mathcal{P}(\mathbb{N}_0)$，其相关函数$f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$由$f(uv)=f(u)+f(v), uv\in E(G)$定义，其中$\mathbb{N}_0$是所有非负整数的集合，$\mathcal{P}(\mathbb{N}_0)$是它的幂集。整数加性集合索引器是整数加性集合标记，使得由$f^+ (uv) = f(u)+ f(v)$定义的导出函数$f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$也是单射的。本文将图的拓扑集标号的概念推广到图的拓扑整数可加集标号。
Local Distance Antimagic Labeling of Neighborhood Balanced Colored Graphs
本文针对无孤立顶点的图定义局部距离反幻标号及相应色数 χ_ld(G)，并研究邻域平衡着色图的标号问题。
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 $w_f(u)\ne w_f(v)$…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。