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Journal of Combinatorial Optimization

, Volume 25, Issue 4, pp 639–645 | Cite as

Three conjectures on the signed cycle domination in graphs

  • Jian Guan
  • Xiaoyan Liu
  • Changhong Lu
  • Zhengke Miao
Article

Abstract

Let G=(V,E) be a graph, a function g:E→{−1,1} is said to be a signed cycle dominating function (SCDF for short) of G if ∑ eE(C) g(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as γ sc (G)=min{∑ eE(G) g(e)∣g is an SCDF of G}. Xu (Discrete Math. 309:1007–1012, 2009) first researched the signed cycle domination number of graphs and raised the following conjectures: (1) Let G be a maximal planar graphs of order n≥3. Then γ sc (G)=n−2; (2) For any graph G with δ(G)=3, γ sc (G)≥1; (3) For any 2-connected graph G, γ sc (G)≥1. In this paper, we present some results about these conjectures.

Keywords

Domination number Signed cycle domination number Planar graph Maximal planar graph 

Notes

Acknowledgements

The first author is supported in part by the National Natural Science Foundation of China (Nos. 70702016 and 70921001); the third author is supported in part by the National Natural Science Foundation of China (Nos. 10971248 and 61073198) and the Fundamental Research Funds for the Central Universities and Program for New Century Excellent Talents in University. The fourth author is supported in part by the National Natural Science Foundation of China (No. 11171288).

References

  1. Diestel R (2006) Graph theory. Springer, Berlin Google Scholar
  2. West DB (1996) Introduction to graph theory. Prentice Hall, Upper Saddle River MATHGoogle Scholar
  3. Xu B (2009) On signed cycle domination in graphs. Discrete Math 309:1007–1012 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jian Guan
    • 1
  • Xiaoyan Liu
    • 2
  • Changhong Lu
    • 2
  • Zhengke Miao
    • 3
  1. 1.Business SchoolCentral South UniversityChangshaChina
  2. 2.Department of MathematicsEast China Normal UniversityShanghaiChina
  3. 3.Department of MathematicsJiangsu Normal UniversityXuzhouChina

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