Antimagic Labelings of Weighted and Oriented Graphs

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.Comment: 10 pages, 1 figur

Regular graphs of odd degree are antimagic

An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield and Ringel conjectured that every connected graph other than the single edge $K_2$ has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.Comment: 5 page

Antimagic Labelings of Caterpillars

A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic.Comment: 13 pages, 4 figure

On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

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 $2C_3$