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Some results on exclusive sum graphs

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Abstract

Let N(Z) denote the set of all positive integers (integers). The sum graph G +(S) of a finite subset S⊂N(Z) is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S⊂N(Z). A sum labelling S is called an exclusive sum labelling if u+v∈S∖V(G) for any edge uv∈E(G). We say that G is labeled exclusively. The least number r of isolated vertices such that G∪rK 1 is an exclusive sum graph is called the exclusive sum number ε(G) of graph G. In this paper, we discuss the exclusive sum number of disjoint union of two graphs and the exclusive sum number of some graph classes.

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References

  1. Harary, F.: Sum graphs and difference graphs. Congr. Numer. 72, 101–108 (1990)

    MathSciNet  Google Scholar 

  2. Harary, F.: Sum graphs over all the integers. Discrete Math. 124, 99–105 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ellingham, M.N.: Sum graphs from trees. Ars Comb. 35, 335–349 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Chen, Z.: Harary’s conjectures on integral sum graphs. Discrete Math. 160, 241–244 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. He, W., Shen, Y., Wang, L., Chang, Y., Kang, Q., Yu, X.: The integral sum number of complete bipartite graphs K r,s. Discrete Math. 239, 137–146 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hartsfield, N., Smyth, W.F.: A family of sparse graphs of large sum number. Discrete Math. 141, 163–171 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. He, W., Yu, X., Mi, H., Xu, Y., Sheng, Y., Wang, L.: The (integral) sum number of K n −E(K r ). Discrete Math. 243, 241–252 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liaw, S., Kuo, D., Chang, G.J.: Integral sum numbers of graphs. Ars Comb. 54, 259–268 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Melnikov, L.S., Pyatkin, A.V.: Regular integral sum graphs. Discrete Math. 252, 237–245 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Miller, M., Ryan, J., Slamin: Integral sum numbers of the cocktail party graphs and symmetric complete bipartite graphs. Bull. ICA 25, 23–28 (1999)

    MATH  MathSciNet  Google Scholar 

  11. Pyatkin, A.V.: New formula for the sum number for the complete bipartite graphs. Discrete Math. 239, 155–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Miller, M., Ryan, J., Slamin, Sugeng, K., Tuga, M.: Exclusive sum graph labeling. Preprint (2003)

  13. Kratochvil, J., Miller, M., Minh, N.H.: Sum Graph labels—An upper bound and related problems. In: Proceedings of the Twelfth Australasian Workshop on Combinatorial Algorithms, Bandun, Indonesia (2001)

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Authors and Affiliations

  1. Department of Mathematics, Shandong Jiaotong University, Jinan, 250023, People’s Republic of China

    Haitang Wang

  2. School of Mathematics Science, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Ping Li

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  1. Haitang Wang
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  2. Ping Li
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Correspondence to Haitang Wang.

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Wang, H., Li, P. Some results on exclusive sum graphs. J. Appl. Math. Comput. 34, 343–351 (2010). https://doi.org/10.1007/s12190-009-0325-7

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  • DOI: https://doi.org/10.1007/s12190-009-0325-7

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