Edge magic and bimagic harmonious labeling of ladder graphs

Abstract

A graph $G=(V, E)$ with $p$ vertices and $q$ edges is said to be edge magic harmonious if there exists a bijection $f: V \cup E \rightarrow\{1,2,3, \ldots, p+q\}$ such that for each edge $x y$ in $E(G)$, the value of $[(f(x)+f(y))(\bmod q)+f(x y)]$ is equal to the constant $k$, called magic constant. A bijection $f: V \cup E \rightarrow\{1,2,3, \ldots, p+q\}$ is called an edge bimagic harmonious labeling if $[(f(x)+f(y))(\bmod q)+f(x y)]=k_1$ or $k_2$ for each edge $x y$ in $E(G)$, where $k_1$ and $k_2$ are two distinct magic constants. A graph $\mathrm{G}$ is said to be edge bimagic harmonious, if it admits an edge bimagic harmonious labeling. Here we prove that the ladder, double ladder are edge bimagic harmonious graphs and circular ladder, triangular ladder are edge magic and bimagic harmonious graphs.