On the odd harmonious graphs with applications

Abstract

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 is odd harmonious if and only if n=2. The construction ways of odd harmonious graph are given. We prove that the graph ∨ =1 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+∪ =1 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.