Crown graphs and subdivision of ladders are odd graceful

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 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.