On totally antimagic total labeling of complete bipartite graphs
Abolape D. Akwu, Deborah O. A. Ajayi
本文研究了完全二部图的全反幻全标号问题。证明了完全二部图是完全反幻全图。我们还证明了有一个顶点的完全二部图的联是完全反幻全图
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
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Totally Antimagic Total labeling of Ladders, Prisms and Generalised Pertersen graphs
给定一个图$G$，如果所有的边权（分别为顶点权）是两两不同的，则$G$上的全标号称为边反魔术全标号（分别为顶点反魔术全标号）。如果在$G$上的一个标号同时是边反魔全标号和顶点反魔全标号，则称为全反魔全标号。允许完全反幻全标号的图称为完全反幻全图。本文证明了梯子图、棱柱图和广义Pertersen图是完全反幻全图。我们还证明了完全反幻全图的链图是完全反幻全图。
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…
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 …
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。
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。
D-Antimagic Labelings of Oriented 2-Regular Graphs
本文研究有向2-正则图的D-反幻标号，分别在不同距离集大小下刻画了有向圈、单向奇圈、Θ-有向圈及相应2-正则图的D-反幻性质。
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 $w_f(u)\ne w_f(v)$…
D-Antimagic Labelings of Oriented Star Forests
本文研究定向星与星森林的 D-反魔法标号，刻画了定向星的全部可行定向和距离集 D，并给出含同构定向星森林的充要条件及存在性结论。
Antimagic orientations of even regular graphs
有向图$D$的$m$弧标号是从$D$的弧集到$\{1, \ldots, m\}$的双射。如果$D$中没有两个顶点具有相同的顶点和，则$D$的标记是反魔术的，其中用于标记的顶点$u\in V(D)$的顶点和是进入$u$的所有弧的标签的和减去离开$u$的所有弧的标签的和。受Hartsfield和Ringel 1990年关于图的反魔标号猜想的启发，Hefetz，Mütze和Schwartz [On antimagic directed graphs，J Graph Theory 64（2010）219--232]开始了有向图的反魔标号的研究，并证明了每个连通图都允许一个反魔方向，其中，如果$D$具有反魔标记，则图$G$的方向$D$是反魔标记。是否每个不相交的循环联盟都承认反魔法取向，这仍然是个未知数。本文首先通过证明每个$2$ -正则图都有一个反魔定向来肯定地回答这个问题。然后，我们表明，任何整数$d\ge2$，每一个连接，$2d$ -正则图有一个反魔方向。我们的技术是新的。
Constructing the antimagic labelings for double stars union paths on three vertices
本文针对反魔法标号问题，确定了使双星图与3顶点路径$P_3$的不交并为反魔法图的$P_3$的确切数量，还给出了顶点和从1到顶点数的$(1,1)$-反魔法标号实例。
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。
Local Distance Antimagic Vertex Coloring of Graphs
本文引入了图的局部距离反魔幻色数参数，并计算了相关图的局部距离反魔幻色数。
A graph $G=(V,E)$ is strongly antimagic, if there is a bijective mapping $f: E \to \{1,2,\ldots,|E|\}$ such that for any two vertices $u\neq v$, not only $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$ and also $\sum_{e \in E(u)}f(e) < \sum_{e\in E(v)}f(e)$ whenever $°(u)< °(v) $, where $E(u)$ is the set of edges incident to $u$. In this paper, we prove that double spiders, the trees contains exactly two vertices of degree at least 3, are strongly antimagic.
A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\ldots,m\}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u \in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$\ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been …
$E_A$-cordial labeling of graphs and its implications for $A$-antimagic labeling of trees
本文研究有限阿贝尔群下图的$E_A$-亲切标号，给出任意循环群下路径为$E_A$-亲切的充要条件，并证伪了已有文献中关于树的$A^*$-反幻标号的猜想。
Antimagic and product antimagic graphs with pendant edges
本文研究带悬挂边的连通图，证明在所有度至少为3的顶点均为支撑顶点时，图对任意正实数等差数列具有反幻标号，并在最小标号不小于1时具有积反幻标号。
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number…
Antimagic Labelings of Caterpillars
本文研究毛毛虫图的反魔标号，通过构造性方法给出一般 $k$-反魔界，并证明若干叶数、路径结构及素数脊阶条件下的反魔或 1-反魔结论。
For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.
An antimagic labeling of a graph $G$ is a $1-1$ correspondence between the edge set $E(G)$ and $\lbrace 1,2,...,|E(G)|\rbrace$ in which the sum of the labels of edges incident to the distinct vertices are different. The edge corona of any two graphs $G$ and $H$, (denoted by $G$ $\diamond$ $H$) is obtained by joining one copy of $G$ with $|E(G)|$ copies of H such that the end vertices of $i^{th}$ edge of $G$ is adjacent to every vertex in the $i^{th}$ copy of $H$. In this paper, we provide an algorithm to prove that the following graphs admit an antimagic labeling: $-$ $n$-barbell graph $B_n$, …
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 $λ(x)+\sum_{y\in V,…
Distance Antimagic Labeling of Zero-Divisor Graphs
该研究针对零因子图的距离反魔法标号问题，证明了多类由模n整数环零因子图构造的并图、笛卡尔积图满足距离反魔法标号的存在性。
Lattice Grids and Prisms are Antimagic
具有$m$边和$n$顶点的有限无向简单图的\emph{antimagic labeling}是从边集到整数$1,...,m$的双射，使得所有$n$顶点和是两两不同的，其中顶点和是与同一顶点关联的所有边的标号之和.如果一个图有一个反魔法标记，则它被称为\emph{antimagic}。1990年，Hartsfield和Ringel证明了除了$K_2$之外的所有连通图都是反魔图。2004年，N. Alon等人证明了这个猜想对具有最小度$Ω(\log n)$的$n$ -顶点图是成立的.他们还证明了完全部图（不同于$K_2$）和最大度至少为$n-2$的$n$ -顶点图是反幻的.最近，Wang证明了环形网格（两个或多个循环的笛卡儿积）是反幻的。在王的论文中，有两个未解决的问题是关于格格图和棱柱图的反魔性，这两个图分别是两条路的笛卡儿积和一个圈与一条路的笛卡尔积。在这篇文章中，我们证明了这两类图是反魔，通过构造这样的反魔标号。
Anti-magic labeling of regular graphs
一个图$G=(V,E)$是反幻的，如果存在一对一的对应关系$f: E \to \{1,2,\ldots, |E|\}$使得对于任意两个顶点$u,v$，$\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$。已知二部正则图是反幻图，奇数度以上的非二部正则图是反幻图。偶数度的非二部正则图是否都是反魔图一直是一个悬而未决的问题。本文解决了这个问题，证明了所有偶度正则图都是反幻图。该论文由《图论杂志》提交至2014年12月。
Antimagic Labelings of Forests
本文为零和划分方法提供算法表示，并证明每个至多含一个2度顶点的森林都是反魔的。
D-Antimagic Labelings on Oriented Linear Forests
本文引入有向图的 D-反幻标号概念，刻画特定距离集下的 D-反幻路径及距离反幻树、森林，并构造有向线性森林上的标号。
Caterpillars Have Antimagic Orientations
有向图$D$与$m$弧的反魔术标号是从$D$的弧集合到$\{1,\dots,m\}$的双射，使得$D$中的顶点的所有定向顶点和是成对不同的，其中顶点$u$的定向顶点和是进入$u$的所有弧的标号之和减去离开$u$的所有弧的标号之和。Hefetz，Mütze，and Schwartz证明了每个连通图都有一个反魔定向，其中图$G$的反魔定向是图$G$有反魔标记的定向.我们使用建设性的技术来证明，毛毛虫，一个著名的子类的树木，有antimagic的方向。
On $k$-Shifted Antimagic Spider Forests
本文研究了$k$-移位反魔术标号问题，证明了特定蜘蛛森林在特定正负整数$k$范围内均满足$k$-移位反魔术性质。
Bipartite graphs with minimum degree at least 15 are antimagic
本文证明了最小度至少为15的二部图是反魔法的，使用了König定理的推论、避免欧拉分量的特定大小子图的存在性以及一个确保某些顶点和能被3整除而其他不能的标号引理。
Antimagic orientation of biregular bipartite graphs
具有$n$顶点和$m$弧的有向图$D$的反魔术标号是从$D$的弧集合到整数$\{1, \cdots, m\}$的双射，使得所有$n$定向顶点和是成对不同的，其中，有向顶点和是进入该顶点的所有弧的标号之和减去离开该顶点的所有弧的标号之和。无向图$G$被称为具有反魔定向如果$G$具有允许反魔标记的取向。Hefetz，M{ü}tze和Schwartz证明了每个连通的无向图都有反魔定向.在本文中，我们通过证明每个双正则二部图都有一个反幻方向来支持这个猜想。
Let $G$ be a graph with $m$ edges and let $f$ be a bijection from $E(G)$ to $\{1,2, \dots, m\}$. For any vertex $v$, denote by $ϕ_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $ϕ_f(v) \neq ϕ_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an {\it antimagic labeling} of $G$. We call $G$ {\it antimagic} if such a labeling exists. Hartsfield and Ringel in 1991 conjectured that all connected graphs except $P_2$ are antimagic. Denote the disjoint union of graphs $G$ and $H$ by $G \cup H$, and the disjoint union of $t$ copies of $G$ by $tG$. For an antimagi…
Regular graphs of odd degree are antimagic
一个具有$m$边的图$G$的反幻标号是从$E(G)$到$\{1,2,\ldots,m\}$的双射，使得对于所有的顶点$u$和$v$，与$u$关联的边的标号之和不同于与$v$关联的边的标号之和. Hartsfield和Ringel证明了除单边图$K_2$之外的所有连通图都有反魔标号.我们对奇数度的正则图证明了这个猜想。
Cartesian Products of Regular Graphs are Antimagic
具有$m$边和$n$顶点的有限无向简单图的\emph{antimagic labeling}是从边集到整数$1,...,m$的双射，使得所有$n$顶点和是两两不同的，其中顶点和是与同一顶点关联的所有边的标号之和.如果一个图有一个反魔法标记，则它被称为\emph{antimagic}。1990年，Hartsfield和Ringel \cite{HaRi}证明了除了$K_2$之外的所有简单连通图都是反魔图。本文证明了一类新的笛卡尔积图是反幻的。此外，将该结果与\cite{Wan}中关于环形网格（两个圈的笛卡儿积）的反幻性结果相结合，可以证明两个或多个正则图的所有笛卡儿积都是反幻的.
Given a digraph $D$ with $m $ arcs, a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $τ$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz in 2010 initiated th…
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, Mütze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $d\ge2$. Together with known res…
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.
Local Antimagic Coloring of Some Graphs
本文研究若干图族的并、冠积和项链图的局部反魔幻色数，并构造出无穷多个满足局部反魔幻色数等于普通色数的图。
An antimagic labeling of a digraph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\cdots,m\}$ such that all $n$ oriented vertex-sums are pairwise distinct, where the oriented vertex-sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph $G$ admits an antimagic orientation if $G$ has an orientation $D$ such that $D$ has an antimagic labeling. Hefetz, M{ü}tze and Schwartz conjectured every connected graph admits an antimagic orientation. In this paper, we support this conjecture by pro…
Dense graphs are antimagic
具有$m$边和$n$顶点的图的{ \em反魔术标号}是从边集到整数$1,...,m$的双射，使得所有$n$顶点和是两两不同的，其中顶点和是与同一顶点关联的所有边的标号之和.一个图被称为{ \em antimagic}，如果它有一个antimagic标签。林格尔的一个猜想（参见\cite{HaRi}）指出，除了$K_2$之外，所有连通图都是反魔的。我们的主要结果验证了这个猜想的图具有最小度$Ω(\log n)$。证明结合了概率参数与简单的工具，从解析数论和组合技术。我们还证明了完全部图（但$K_2$）和最大度至少为$n-2$的图是反幻的。
Super Total Local Antimagic Vertex Coloring of Graphs
本文定义超顶点与超边全局部反魔法标号及其诱导色数，并计算若干图族的相应色数。
An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph $G$ has an antimagic orientation if it has an orientation which admits an antimagic labeling. Hefetz, M{ü}tze, and Schwartz conjectured that every connected graph admits an antimagic orientation. In this paper, we show that every bipartite grap…
Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a {\bf local antimagic labeling} if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e. $w(u) \neq w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color(vertex sum) $w(v)$. The {\bf local antimagic chromatic number} $χ_{la}(G)$ is the minimum number of colo…
An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label of $x$ is $f^+(x)= \sum_{e\in E(x)} f(e)$ ($E(x)$ is the set of edges incident to $x$). The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graph…
Some Results on Local Distance Antimagic Chromatic Number of Graphs
本文针对不含孤立顶点的图，研究图的联图及图与空图的字典积的局部距离反魔法色数，推进了该图论参数的相关研究。
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number…
Caterpillars are Antimagic
针对 Hartsfield–Ringel 反魔法标号猜想在树类上的开放问题，本文提出一种 $O(n\log n)$ 标号算法，证明所有毛虫图均为反魔法图。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, several sufficient conditions for $χ_{la}(H)\le χ_{la}(G)$ are obtained, where $H$ is obtained from $G$ …
On bridge graphs with local antimagic chromatic number 3
本文研究了局部反魔幻色数为3的桥图，提出了几族满足该条件的桥图，并给出了多种构造局部反魔幻色数为3的桥图的方法。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…
A graph $G = (V, E)$ of order $p$ and size $q$ is said to be local antimagic if there exists a bijection $g:E(G) \to \{1,2,\ldots,q\}$ such that for any pair of adjacent vertices $u$ and $v$, $g^+(u)\ne g^+(v)$, where $g^+(u)=\sum_{uv\in E(G)} g(uv)$ is the induced vertex color of $u$ under $g$. We also say $G$ is local antimagic total if there exists a bijection $f: V\cup E \to\{1,2,\ldots ,p+q\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\not= w(v)$, where $w(u)= f(u) +\sum_{uv\in E(G)} f(uv)$ is the induced vertex weight of $u$ under $f$. The local antimagic (and local a…
Local antimagic chromatic number of partite graphs
本文研究由局部反幻边标号诱导的正常顶点着色，确定了完全三部图 K_{1,m,n} 及奇偶性不同的 K_{m,n} 的 r 个副本的局部反幻色数。
Necessary and sufficient conditions of a class of bipartite graphs with local antimagic chromatic number 2 - an algebraic approach
本文运用代数分析，得出了一类除3个顶点度数≥3外其余顶点度数均为2的二部图的局部反魔法色数为2的充要条件，并证明了对应诱导路径的连续边标号互不相同。
List-antimagic labeling of vertex-weighted graphs
本文证明了任意不含孤立点或孤立边分量的n顶点图是⌊4n/3⌋-加权列表反幻影的，且任意不含孤立点分量的n顶点图存在一个方向使其成为⌊2n/3⌋-有向反幻影的。
Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained…
Trees whose even-degree vertices induce a path are antimagic
针对 Hartsfield–Ringel 反魔法标号猜想在树类上的开放问题，本文证明偶度顶点诱导子图为路径的树均存在反魔法标号，推广了已有结果。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we give counterexamples to the lower bound of $χ_{la}(G \vee O_2)$ that was obtained in [Local antimagic…
On Distance Antimagic Graphs
对于任意距离集$D\subseteq \{0,1, \ldots, diam(G)\}$，图$G$中顶点$x$在顶点标号$f:V\rightarrow \{1,2, \ldots , v\}$下的$D$ -权定义为$w_D(x)=\sum_{y\in N_D(x)} f(y)$，其中$N_D(x) = \{y \in V| d(x,y) \in D\}$ .一个图$G$被称为$D$ -距离幻数，如果所有顶点具有相同的$D$ -顶点-权重;它被称为$D$ -距离反幻数，如果所有顶点具有不同的$D$ -顶点-权重;它被称为$(a,d)-D$ -距离反幻数，如果$D$ -顶点-权重构成一个具有差值$d$和起始值$a$的等差数列.本文研究了$D$ -距离反幻图存在的一些必要条件。我们猜想这样的条件也是充分的。此外，我们还研究了一些与圈相关的连通图的$\{1\}$ -距离反魔术标号：圈，太阳，棱柱，完全图，轮子，风扇，和友谊图。
Antimagic Labeling of Graphs Using Prime Numbers
本文研究图和树的反幻标号，通过为边分配不同素数使顶点处边标签累加和唯一，提出猜想并证明了完美二叉树和完全图的反幻标号可行性。
A {\it local antimagic labeling} of a connected graph $G$ with at least three vertices, is a bijection $f:E(G) \rightarrow \{1,2,\ldots , |E(G)|\}$ such that for any two adjacent vertices $u$ and $v$ of $G$, the condition $ω_{f}(u) \neq ω_{f}(v)$ holds; where $ω_{f}(u)=\sum _{x\in N(u)} f(xu)$. Assigning $ω_{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of $G$, denoted by $χ_{la}(G)$, is defined as the minimum of $|f|$, wh…
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