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An edge labeling of a connected graph $G = (V, E)$ is said to be localantimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for anypair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the inducedvertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incidentto $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$,is the minimum number of distinct induced vertex labels over all localantimagic labelings of $G$. In this paper, we (i) give a sufficient conditionfor a graph with one pendant to have $\chi_{la}\ge 3$. A necessary andsufficient condition for a graph to have $\chi_{la}=2$ is then obtained; (ii)give a sufficient condition for every circulant graph of even order to have$\chi_{la} = 3$; (iii) construct infinitely many bipartite and tripartitegraphs with $\chi_{la} = 3$ by transformation of cycles; (iv) applytransformation of cycles to obtain infinitely many one-point union of regular(possibly circulant) or bi-regular graphs with $\chi_{la} = 2,3$. The work ofthis paper suggests many open problems on the local antimagic chromatic numberof bipartite and tripartite graphs.

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number