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 $f:E \to\{1,\ldots ,|E|\}$ 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 $G$. 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 $\chi_{la}(T_k) = k+1$.