On number of pendants in 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 $G$. Let $\chi (G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of ${\chi }_{la}(G)$ for $G$ with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for $k\ge 1$, there are infinitely many graphs with $k\ge \chi (G)-1$ pendant vertices and ${\chi }_{la}(G)=k+1$. We conjecture that every tree $T_k$, other than certain caterpillars, spiders and lobsters, with $k\ge 1$ pendant vertices has ${\chi }_{la}(T_k)=k+1$.

6 page, 3 figures, a new short paper that gives tight upper and lower bounds with sufficient conditions for the bounds to be equal