In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0
or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved
that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd
harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by
two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show
that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ ,2 m ≡ 0 (mod )4 are odd
harmonious graphs. Finally many new families of odd harmonious graphs are introduced. |
