Odd Harmonious and Strongly Odd Harmonious Graphs

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 $f:V(G){\rightarrow}\{0,\;1,\;2,\;{\ldots},\;2m-1\}$ such that the induced function $f^*:E(G){\rightarrow}\{1,\;3,\;5,\;{\ldots},\;2m-1\}$ 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 $f_A:A{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ and $f_B:B{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ such that the induced labeling on the edges $f_{E(G)}:E(G){\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ defined by $f_{E(G)}(uv)=f_A(u)-f_B(v)$ (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.