On local antimagic chromatic number of graphs with cut-vertices

An edge labeling of a connected graph $G=(V,E)$ is said to be local antimagic if it is a bijection $f:E\to \{1,\dots ,\mathrm{\mid }E\mathrm{\mid }\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^{+}(x)≠f^{+}(y)$, where the induced vertex label $f^{+}(x)=\sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by ${\chi }_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. The exact value of the local antimagic chromatic number of many families of graphs with cut-vertices (possibly given by pendant edges) are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, {\it Graphs and Combin.}, {\bf33} (2017), 275–285.].