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

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

Local distance irregular labeling of graphs

We introduce the notion of distance irregular labeling, called the local distance irregular labeling. We define λ : V (G) −→ {1, 2, . . . , k} such that the weight calculated at the vertices induces a vertex coloring if w(u) 6≠ w(v) for any edge uv. The weight of a vertex u ∈ V (G) is defined as the sum of the labels of all vertices adjacent to u (distance 1 from u), that is w(u) = Σy∈N(u)λ(y). The minimum cardinality of the largest label over all such irregular assignment is called the local distance irregularity strength, denoted by disl(G). In this paper, we found the lower bound of the local distance irregularity strength of graphs G and also exact values of some classes of graphs namely path, cycle, star graph, complete graph, (n, m)-tadpole graph, unicycle with two pendant, binary tree graph, complete bipartite graphs, sun graph.Publisher's Versio

Approximate results for rainbow labelings

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ON SUPER (3n+5,2)- EDGE ANTIMAGIC TOTAL LABELING AND IT’S APPLICATION TO CONSTRUCT HILL CHIPER ALGORITHM

Graph labeling can be implemented in solving problems for various fields of life.  One of the application of graph labelling is in security system. Information security is needed to reduce risk, data manipulation, and unauthorized destruction or destruction of information. Cryptographic algorithms that can be used to build security systems, one of the cryptographic algorithms is Hill Cipher. Hill chipper is a cryptographic algorithm that uses a matrix as a key to perform encryption, decryption, and modulo arithmetic. This study discusses the use of Super (3n+5,2)- edge antimagic total labeling to construct the Hill Chiper algorithm. The variation of the edge weight function and the corresponding edge label on the  graph, will make the constructed lock more complicated to hac

Approximate results for rainbow labelings

The final publication is available at Springer via https://doi.org/10.1007/s10998-016-0151-2]A simple graph G=(V,E) is said to be antimagic if there exists a bijection f:E¿[1,|E|] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f:V¿[1,|V|], such that ¿x,y¿V, ¿xi¿N(x)f(xi)¿¿xj¿N(y)f(xj). Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval [1,2n+m-4] and, for trees with k inner vertices, in the interval [1,m+k]. In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree ¿ in the interval [1,n+t(n-t)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n-4k]. In particular, all trees with n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.Peer ReviewedPostprint (author's final draft

Magic and antimagic labeling of graphs

"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."Doctor of Philosoph