On local antimagic chromatic number of cycle-related join graphs

Abstract

An edge labeling of a connected graph $G=(V,E)$ is said to be local antimagic if it is a bijection $f:E\to \{1,\dots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^{+}(x)\ne f^{+}(y)$, 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, several sufficient conditions for ${\chi }_{la}(H)\le {\chi }_{la}(G)$ are obtained, where $H$ is obtained from $G$ with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.