On bridge graphs with local antimagic chromatic number 3

Abstract

Let $G=(V,E)$ be a connected graph. A bijection $f:E\to \{1,\dots ,|E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^{+}(x)\ne f^{+}(y)$, where $f^{+}(x)={\sum }_{e\in E(x)}f(e)$ and $E(x)$ is the set of edges incident to $x$. Thus a local antimagic labeling induces a proper vertex coloring of $G$, where the vertex $x$ is assigned the color $f^{+}(x)$. The local antimagic chromatic number ${\chi }_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. In this paper, we present some families of bridge graphs with ${\chi }_{la}(G)=3$ and give several ways to construct bridge graphs with ${\chi }_{la}(G)=3$.