Some Cartesian Products of a Path and Prism Related Graphs That Are Edge Odd Graceful

Abstract

Let $G$ be a connected undirected simple graph of size $q$ and let $k$ be the maximum number of its order and its size. Let $f$ be a bijective edge labeling which codomain is the set of odd integers from 1 up to $2q-1$. Then $f$ is called an edge odd graceful on $G$ if the weights of all vertices are distinct, where the weight of a vertex $v$ is defined as the sum $mod(2k)$ of all labels of edges incident to $v$. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs.