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Local Antimagic Vertex Coloring of Unicyclic Graphs

Last modified: 2017-07-05

#### Abstract

The local antimagic labeling on a graph $G$with $n$ vertices and $m$ edges is defined to be an assignment $f : E \rightarrow \{1, 2, \cdots, m\}$ so that the weights of any two adjacent vertices $u$ and $v$ are distinct, that is, $w(u) \neq \w(v)$ where $w(u) = \Sigma_{e \in E(u)} f(e)$and $E(u)$ is the set of edges incident to $u$. Therefore, any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $u$ is assigned the color $w(u)$. The local antimagic chromatic number, denoted by $\chi_{la}(G)$, is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and sun.