# On the odd harmonious graphs with applications

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## 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## Mathematics Subject Classification (2000)

05C78## Preview

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## References

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