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

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

Approximate results for rainbow labelings

Article de recercaPreprin

Weighted-1-antimagic graphs of prime power order

AbstractSuppose G is a graph, k is a non-negative integer. We say G is weighted-k-antimagic if for any vertex weight function w:V→N, there is an injection f:E→{1,2,…,∣E∣+k} such that for any two distinct vertices u and v, ∑e∈E(v)f(e)+w(v)≠∑e∈E(u)f(e)+w(u). There are connected graphs G≠K2 which are not weighted-1-antimagic. It was asked in Wong and Zhu (in press) [13] whether every connected graph other than K2 is weighted-2-antimagic, and whether every connected graph on an odd number of vertices is weighted-1-antimagic. It was proved in Wong and Zhu (in press) [13] that if a connected graph G has a universal vertex, then G is weighted-2-antimagic, and moreover if G has an odd number of vertices, then G is weighted-1-antimagic. In this paper, by restricting to graphs of odd prime power order, we improve this result in two directions: if G has odd prime power order pz and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of p, then G is weighted-1-antimagic. If G has odd prime power order pz, p≠3 and has maximum degree at least ∣V(G)∣−3, then G is weighted-1-antimagic

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

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