Trees whose even-degree vertices induce a path are antimagic

An antimagic labeling of a connected graph G is a bijection from the set of edges E(G) to {1, 2, . . . , |E(G)|} such that all vertex sums are pairwise distinct, where the vertex sum at vertex v is the sum of the labels assigned to edges incident to v. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than K2 is antimagic; however the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Anti-magic labeling of trees, Discrete Math. 331 (2014) 9–14].A. Lozano is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement ERC-2014-CoG 648276 AUTAR); M. Mora is supported by projects Gen. Cat. DGR 2017SGR1336, MINECO MTM2015-63791-R, and H2020-MSCARISE project 734922-CONNECT; and C. Seara is supported by projects Gen. Cat. DGR 2017SGR1640, MINECO MTM2015-63791-R, and H2020-MSCARISE project 734922-CONNECT.Peer ReviewedPostprint (published version

Trees whose even-degree vertices induce a path are antimagic

An antimagic labeling of a connected graph G is a bijection from the set of edges E(G) to {1, 2, . . . , |E(G)|} such that all vertex sums are pairwise distinct, where the vertex sum at vertex v is the sum of the labels assigned to edges incident to v. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than K2 is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9–14].Preprin

An Inductive Approach to Strongly Antimagic Labelings of Graphs

An antimagic labeling for a graph $G$ with $m$ edges is a bijection $f: E(G) \to \{1, 2, \dots, m\}$ so that $\phi_f(u) \neq \phi_f(v)$ holds for any pair of distinct vertices $u, v \in V(G)$, where $\phi_f(x) = \sum_{x \in e} f(e)$. A strongly antimagic labeling is an antimagic labeling with an additional condition: For any $u, v \in V(G)$, if $\deg(u) > \deg(v)$, then $\phi_f(u) > \phi_f(v)$. 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 spiders are strongly antimagic, previously shown by Shang [Spiders are antimagic, Ars Combinatoria, 118 (2015), 367--372] and Huang [Antimagic labeling on spiders, Master's Thesis, Department of Mathematics, National Taiwan University, 2015], and by Chang, Chin, Li and Pan [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68], respectively. We fix a subtle error in [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68]. Further, we prove certain level-wise regular trees, cycle spiders and cycle double spiders are all strongly antimagic

Antimagic Labeling for Unions of Graphs with Many Three-Paths

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 $\phi_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $\phi_f(v) \neq \phi_f(u)$ 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 $\tau(G)$ to be the maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq \tau(G)$. Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $\tau(G)$ 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 $\tau(G)$ 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 $\tau(G)$ for these special families. In the present paper, we generalize their results by proving an upper bound on $\tau(G)$ 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 $2C_3$

Shifted-Antimagic Labelings for Graphs

[[abstract]]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 the literature of antimagic labelings and extends previous results in three aspects: -Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be k-shifted-antimagic for sufficiently large k. -Some graphs are proved k-shifted-antimagic for all k, while some are proved not for some particular k. -Disconnected graphs are also considered.[[notice]]補正完

Antimagic Labeling on Spiders

設圖G 是一由n 個點及m 條邊組成的有限簡單圖，圖G 的一個標號指的是在圖G 的每一個邊標上一個{1, 2, · · · ,m} 內的整數，且不同邊有不同標號。給定圖G 一個標號，定義每個頂點的頂點和是這個點所有連出去的邊的標號總和，若圖G 所有頂點的頂點和都不一樣，則稱此標號為反魔方標號；設f 是圖G 的一個反魔方標號，且對於任兩個度數不同的頂點u, v, deg(u) < deg(v)，若u 的頂點和嚴格小於v 的頂點和，則稱f 是圖G 的一個強反魔方標號。另外，若圖G 存在一個(強) 反魔方標號，我們稱G 是(強) 反魔方的。 反魔方標號一詞最早是由Hartsfield 和Ringel 提出，在他們的著作裡不只證明幾個簡單的例子(圈、路徑、輪子、完全圖等) 有反魔方標號，也同時提出所有不是K2 的連通圖都是反魔方的猜想。幾十年來， 陸陸續續有人證明滿足某些條件的圖有反魔方標號，但距離此猜想完全解決仍有很大的空間。 在本篇論文中，我們將範圍限縮到蜘蛛圖(有一個核心和至少三隻腳，每隻腳由數條邊組成)。由於這種圖已被證實具有反魔方標號，因此我們在這裡將證明一個更強的結果：所有的蜘蛛圖都有強反魔方標號。文章最後也會討論一些蜘蛛圖的變形是反魔方的。Let G be a simple finite graph with n vertices and m edges. A labeling of G is a bijection from the set of edges to the set {1, 2, · · · ,m} of integers. Given a labeling of G, for each vertex, its vertex sum is defined to be the sum of labels of all edges incident to it. If all vertices have distinct vertex sums, we call this labeling antimagic. Suppose f is an antimagic labeling of G, and for any two vertices u, v with deg(u) < deg(v), if vertex sum of u is strictly less than vertex sum of v, then we say f is a strongly antimagic labeling of G. Furthermore, a graph G is said to be (strongly) antimagic if it has (a strongly) an antimagic labeling. The concept of antimagic labeling was first introduced by Hartsfield and Ringel. In their book, they not only proved that some graphs such as cycles, paths, wheels, complete graphs etc are antimagic, but also conjectured that all connected graphs other than K2 are antimagic. In the past years, graphs with some restriction were gradually poven to be antimagic, but this conjecture is still widely open. In this thesis, we restrict our graphs to spiders, which is a graph with a core and at least three legs, each leg contains some edges. Since all spiders have already been proven to be antimagic, we will prove a stronger result here, that is, all spiders are strongly antimagic. In the last chapter, we will discuss whether some variation of spiders are antimagic or not