76 research outputs found
Local Antimagic Coloring of Some Graphs
Given a graph , a bijection
is called a local antimagic labeling of if the vertex weight is distinct for all adjacent vertices. The vertex
weights under the local antimagic labeling of induce a proper vertex
coloring of a graph . The \textit{local antimagic chromatic number} of
denoted by is the minimum number of weights taken over all such
local antimagic labelings of . In this paper, we investigate the local
antimagic chromatic numbers of the union of some families of graphs, corona
product of graphs, and necklace graph and we construct infinitely many graphs
satisfying
Lattice Grids and Prisms are Antimagic
An \emph{antimagic labeling} of a finite undirected simple graph with
edges and vertices is a bijection from the set of edges to the integers
such that all 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 , is antimagic. In
2004, N. Alon et al showed that this conjecture is true for -vertex graphs
with minimum degree . They also proved that complete partite
graphs (other than ) and -vertex graphs with maximum degree at least
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
Complete characterization of s-bridge graphs with local antimagic chromatic number 2
An edge labeling of a connected graph is said to be local
antimagic if it is a bijection such that for any
pair of adjacent vertices and , , where the induced
vertex label , with ranging over all the edges incident
to . The local antimagic chromatic number of , denoted by ,
is the minimum number of distinct induced vertex labels over all local
antimagic labelings of . In this paper, we characterize -bridge graphs
with local antimagic chromatic number 2
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