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On the local edge Antimagicness of $m$-splitting Graphs

Last modified: 2017-07-13

#### Abstract

Let $G$ be a connected and simple graph. A split graph is a graph derived by adding new vertex $v'$ in every vertex $v$ such that $v'$ adjacent to $v$ in graph $G$. An $m$-splitting graph is a graph which has $m$ $v'$-vertices, denoted by $_{m}Spl(G)$. A local edge antimagic coloring in $G=(V,E)$ graph is a bijection $f:V(G)\longrightarrow \{1,2,3,...,|V(G)|\}$ in which for any two adjacent edges $e_1$ and $e_2$ satisfies $w(e_1)\neq w(e_2)$, where $e=uv\in G$. The color of any edge $e=uv$ are assigned by $w(e)$ which is defined by sum of label both end vertices $f(u)$ and $f(v)$. The chromatic number of local edge antimagic labeling $\gamma_{lea}(G)$ is the minimal number of color of edge in $G$ graph which has local antimagic coloring. We present the exact value of chromatic number $\gamma_{lea}$ of $m$-splitting graph and study its bound.