On the odd harmonious graphs with applications

The necessary conditions for the existence of odd harmonious labelling of graph are obtained. A cycle C _n is odd harmonious if and only if n≡0 (mod 4). A complete graph K _n is odd harmonious if and only if n=2. A complete k-partite graph K(n ₁,n ₂,…,n _k ) is odd harmonious if and only if k=2. A windmill graph K _n ^t is odd harmonious if and only if n=2. The construction ways of odd harmonious graph are given. We prove that the graph ∨ _i=1 ⁿ G _i , the graph G(+r ₁,+r ₂,…,+r _p ), the graph $\bar{K_{m}}+_{0}P_{n}+_{e}\bar{K_{t}}$ , the graph G∪(X+∪ _k=1 ⁿ Y _k ), some trees and the product graph P _m ×P _n etc. are odd harmonious. The odd harmoniousness of graph can be used to solve undetermined equation.