Kyungpook Mathematical Journal 2018; 58(4): 747-759
Published online December 31, 2018
Copyright © Kyungpook Mathematical Journal.
Odd Harmonious and Strongly Odd Harmonious Graphs
Mohamed Abdel-Azim Seoud, Hamdy Mohamed Hafez∗
Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt
e-mail : m.a.seoud@hotmail.com
Department of Basic science, Faculty of Computers and Information, Fayoum University, Fayoum 63514, Egypt
e-mail : hha00@fayoum.edu.eg
Received: April 12, 2017; Revised: August 12, 2018; Accepted: October 2, 2018
A graph G = (V (G), E(G) of order n = |V (G)| and size m = |E(G)| is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, …, 2m−1} such that the induced function f* : E(G) → {1, 3, 5, …, 2m−1} defined by f*(uv) = f(u)+f(v) is bijection. While a bipartite graph G with partite sets A and B is said to be bigraceful if there exist a pair of injective functions fA : A → {0, 1, …, m − 1} and fB : B → {0, 1, …, m − 1} such that the induced labeling on the edges fE(G) : E(G) → {0, 1, …, m − 1} defined by fE(G)(uv) = fA(u)−fB(v) (with respect to the ordered partition (A, B)), is also injective. In this paper we prove that odd harmonious graphs and bigraceful graphs are equivalent. We also prove that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is ⌈m/2⌉! ⌊m/2⌋!. We prove that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious. We find some new disconnected odd harmonious graphs.
Keywords: odd harmonious graphs, labeling, cartesian product.