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

Lattice Grids and Prisms are Antimagic

An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but $K_2$, is antimagic. In 2004, N. Alon et al showed that this conjecture is true for $n$-vertex graphs with minimum degree $\Omega(\log n)$. They also proved that complete partite graphs (other than $K_2$) and $n$-vertex graphs with maximum degree at least $n-2$ are antimagic. Recently, Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang's paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labelings.Comment: 10 pages, 6 figure

The Integer-antimagic Spectra of Graphs with a Chord

Let $A$ be a nontrival abelian group. A connected simple graph $G = (V, E)$ is $A$-antimagic if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \sum_{uv\in E(G)}f(uv)$, is injective. The integer-antimagic spectrum of a graph $G$ is the set IAM$(G) = \{k\;|\; G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic}$ $\textnormal{and } k \geq 2\}$. In this paper, we determine the integer-antimagic spectra for cycles with a chord, paths with a chord, and wheels with a chord

Group-antimagic Labelings of Multi-cyclic Graphs

Let $A$ be a non-trivial abelian group. A connected simple graph $G = (V, E)$ is $A$-\textbf{antimagic} if there exists an edge labeling $f: E(G) \to A \backslash \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \Sigma$ $\{f(u,v): (u, v) \in E(G) \}$, is a one-to-one map. The \textit{integer-antimagic spectrum} of a graph $G$ is the set IAM$(G) = \{k: G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic and } k \geq 2\}$. In this paper, we analyze the integer-antimagic spectra for various classes of multi-cyclic graphs

An Example Usage of Graph Theory in Other Scientific Fields: On Graph Labeling, Possibilities and Role of Mind/Consciousness

This paper provides insights into some aspects of the possibilities and role of mind, consciousness, and their relation to mathematical logic with the application of problem solving in the fields of psychology and graph theory. This work aims to dispel certain long-held notions of a severe psychological disorder and a well-known graph labeling conjecture. The applications of graph labelings of various types for various kinds of graphs are being discussed. Certain results in graph labelings using computer software are presented with a direction to discover more applications

Antimagic Labeling of Some Degree Splitting Graphs

A graph with q edges is called antimagic if its edges can be labeled with 1, 2, 3, ..., q without repetition such that the sums of the labels of the edges incident to each vertex are distinct.  As Wang et al. [2012], proved that not all graphs are antimagic, it is interesting to investigate antimagic labeling of graph families. In this paper we discussed antimagic labeling of the larger graphs obtained using degree splitting operation on some known antimagic graphs.  As discussed in Krishnaa [2016], antimagic labeling has many applications, our results will be used to expand the network on larger graphs