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

Abstract

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, 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, we (i) give a sufficient condition for a graph with one pendant to have $\chi_{la}\ge 3$. A necessary and sufficient 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 tripartite graphs with $\chi_{la} = 3$ by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with $\chi_{la} = 2,3$. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.