On Integer-Magic Spectra of Caterpillars

Abstract

For any $h\in \mathbb{N}$, a graph $G=(V,E)$ is said to be $h$-magic if there exists a labeling $l:E(G)\to {\mathbb{Z}}_h–\{0\}$ such that the induced vertex set labeling $l^{+}:V(G)\to {\mathbb{Z}}_h$ defined by

$$l^{+}(v)=\sum _{uv\in E(G)}l(uv)$$

is a constant map. For a given graph $G$, the set of all $h\in {\mathbb{Z}}_{+}$ for which $G$ is $h$-magic is called the integer-magic spectrum of $G$ and is denoted by $IM(G)$. The concept of integer-magic spectrum of a graph was first introduced in [4]. But unfortunately, this paper has a number of incorrect statements and theorems. In this paper, first we will correct some of those statements, then we will determine the integer-magic spectra of caterpillars.