Antimagic orientation of lobsters

A note on antimagic orientations of even regular graphs

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…

Antimagic orientations of disconnected even regular graphs

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 …

Antimagic orientations of graphs with large maximum degree

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…

含支配团的图的反魔法定向

Antimagic orientations of graphs with a dominating clique

本文针对含支配团的图，证明了Hefetz等人提出的“每个连通图都存在反魔法定向”的猜想成立，完善了反魔法定向领域的相关结论。

完全图的反魔法标号

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。

On number of pendants in local antimagic chromatic number

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 $G$. Let $χ(G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of $χ_{la}(G)$ for $G$ with pend…

The antimagic orientation problems for graphs obtained by some graph operations

A simple graph $G$ is said to admit an antimagic orientation if there exist an orientation on the edges of $G$ and a bijection from $E(G)$ to $\{1,2,\ldots,|E(G)|\}$ such that the vertex sums of vertices are pairwise distinct, where the vertex sum of a vertex is defined to be the sum of the labels of the in-edges minus that of the out-edges incident to the vertex. It was conjectured by Hefetz, Mütze, and Schwartz~\cite{HMS10} in 2010 that every connected simple graph admits an antimagic orientation. In this paper, we prove that the Mycielski construction and the corona product for graphs with …

Antimagic orientations of graphs with given independence number

Given a digraph $D$ with $m$ arcs and a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$, we say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $τ$, 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 $u$. Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few …

Antimagic orientation of subdivided caterpillars

Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $τ:A(D)\rightarrow \{1,2,\ldots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $τ$, where the vertex-sum of a vertex $v$ in $D$ under $τ$ is the sum of labels of all arcs entering $v$ minus the sum of labels of all arcs leaving $v$. Hefetz, Mütze and Schwartz [J. Graph Theory, 64: 219-232, 2010] conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of grap…

定向星森林的 D-反魔法标号

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。

每个三次图的2-细分都是反魔术图

Every 2-Subdivision of a Cubic Graph Is Antimagic

针对三次图2-细分的反魔术图问题，通过两种构造方法证明其为反魔术图，且当原三次图顶点度数为≥3的奇数时为强反魔术图，补充了Li(2025)方法未覆盖的三次图情况。

顶点加权图的反幻影标号

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。