Journal of Applied Mathematics and Computing

, Volume 29, Issue 1–2, pp 105–116

# On the odd harmonious graphs with applications

• Zhi-He Liang
• Zhan-Li Bai
Article

## 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 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.

## Keywords

Odd harmonious graph Harmonious graph Tree Product graph

05C78

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