Pell graceful labeling of graphs

Abstract

In this paper, we introduce a new concept of Pell graceful labeling as follows. An injective function $f$ from $V(G)$ into $\left\{0,1,2, \ldots, p_q\right\}$ is Pell graceful if the induced edge labeling $f^*(u v)=|f(u)-f(v)|$ is a bijection onto the set $\left\{p_1, p_2, \ldots, p_q\right\}$. A graph $G(p, q)$ which admits a Pell graceful labeling is called a Pell graceful graph, where $p_q$ is the $q^{\text {th }}$ Pell number in the Pell sequence. Here, Pell graceful labeling of some family of graphs are obtained. Its non-existence are established.