Advertisement

Journal of Applied Mathematics and Computing

, Volume 29, Issue 1–2, pp 105–116 | Cite as

On the odd harmonious graphs with applications

  • Zhi-He LiangEmail author
  • 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 

Mathematics Subject Classification (2000)

05C78 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Graham, R.L., Sloane, N.J.A.: On additive bases and harmonious graphs. SIAM J. Algebr. Discrete Methods 1, 382–404 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Liu, B.: Strongly harmonious equation of graph with application. J. Xinjiang Univ. 1, 30–34 (1988) Google Scholar
  3. 3.
    Liang, Z.: Harmoniousness of quadrilateral cactus. J. Eng. Math. 1, 131–134 (2001) Google Scholar
  4. 4.
    Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Comb. 5, 1–148 (2005), DS#6 MathSciNetGoogle Scholar
  5. 5.
    Youssef, M.Z.: Two general results on harmonious labeling. Ars Comb. 68, 225–230 (2003) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Lu, H.-C.: On the constructions of sequential graphs. Taiwan. J. Math. 4, 1095–1107 (2006) Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2008

Authors and Affiliations

  1. 1.Department of MathematicsHebei Normal UniversityShijiazhuangPeople’s Republic of China
  2. 2.Editorial Department of Hebei Normal University JournalShijiazhuangPeople’s Republic of China

Personalised recommendations