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 $f:E\to \{1,\dots ,|E|\}$ 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, 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.

arXiv commentA work that produces infinitely many bipartite graphs with local antimagic chromatic number is 2 or 3, and infinitely many tripartite graphs with local antimagic chromatic number is 3. Many open problems on bipartite and tripartite graphs are suggested

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