Crown graphs and subdivision of ladders are odd graceful

Abstract

A graph G of size q is odd graceful, if there is an injection φ from V(G) to {0, 1, 2, …, 2q−1} such that, when each edge xy is assigned the label or weight |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, …, 2q−1}. This definition was introduced in 1991 by Gnanajothi [3], who proved that the graphs obtained by joining a single pendant edge to each vertex of C n are odd graceful, if n is even. In this paper, we generalize Gnanajothi's result on cycles by showing that the graphs obtained by joining m pendant edges to each vertex of C n are odd graceful if n is even. We also prove that the subdivision of ladders S(L n ) (the graphs obtained by subdividing every edge of L n exactly once) is odd graceful.