论文标题
具有许多三路径图的工会的Antagic标记
Antimagic Labeling for Unions of Graphs with Many Three-Paths
论文作者
论文摘要
令$ g $是带有$ m $边缘的图形，让$ f $是$ e（g）$到$ \ {1,2，\ dots，m \} $的双线。对于任何顶点$ v $，用$ ϕ_f（v）$表示$ f（e）$的$ e $ $ e $事件的总和为$ v $。如果$ ϕ_f（v）\ neq ϕ_f（u）$对于任何两个不同的顶点$ u $和$ v $保留，则$ f $称为$ g $的{\ it antimagic labeling}。如果存在这样的标签，我们称$ g $ {\ it Antimagic}。 Hartsfield和Ringel在1991年猜想，除$ P_2 $以外的所有连接图都是Antimagic。表示$ g \ cup h $的图形$ g $和$ h $的不连接联盟，以及$ g $ $ g $ by $ tg $的$ t $副本的分离。对于Antimagic Graph $ g $（连接或断开连接），我们定义参数$τ（g）$是最大整数，以使$ g \ cup tp_3 $对于所有$ t \ leqτ（g）$都是抗原。 Chang，Chen，Li和Pan表明，对于所有抗原图，$ g $，$τ（g）$都是有限的[Graphs and Combinatorics 37（2021），1065---1182]。此外，Shang，Lin，Liaw [Util。数学。 97（2015），373--385]和Li [台湾国家钟兴大学的硕士论文，2019年]，分别为特殊的图形家庭找到了$τ（g）$的确切价值：星际森林和平衡的双星星。他们通过寻找$ g \ cup tp_3 $的显式抗原标签来做到这一点，并证明了这些特殊家庭的$τ（g）$上的紧密上限。在本文中，我们通过证明所有图表上的$τ（g）$上的上限来概括他们的结果。对于星森林和平衡的双星星，这种一般界限等同于\ cite {Star Forest}和\ Cite {double Star}中给出的边界。此外，我们证明了我们研究过的所有其他图表的一般界限也很紧，包括无限的水母图，循环$ C_N $，其中$ 3 \ leq n \ leq n \ leq 9 $和双三角$ 2C_3 $。
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 antimagic graph $G$ (connected or disconnected), we define the parameter $τ(G)$ to be the maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq τ(G)$. Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $τ(G)$ is finite [Graphs and Combinatorics 37 (2021), 1065--1182]. Further, Shang, Lin, Liaw [Util. Math. 97 (2015), 373--385] and Li [Master Thesis, National Chung Hsing University, Taiwan, 2019] found the exact value of $τ(G)$ for special families of graphs: star forests and balanced double stars respectively. They did this by finding explicit antimagic labelings of $G\cup tP_3$ and proving a tight upper bound on $τ(G)$ for these special families. In the present paper, we generalize their results by proving an upper bound on $τ(G)$ for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in \cite{star forest} and \cite{double star} and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles $C_n$ where $3 \leq n \leq 9$, and the double triangle $2C_3$.
