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 ∑ e∈E(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{∑ e∈E(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.
Similar content being viewed by others
References
Diestel R (2006) Graph theory. Springer, Berlin
West DB (1996) Introduction to graph theory. Prentice Hall, Upper Saddle River
Xu B (2009) On signed cycle domination in graphs. Discrete Math 309:1007–1012
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guan, J., Liu, X., Lu, C. et al. Three conjectures on the signed cycle domination in graphs. J Comb Optim 25, 639–645 (2013). https://doi.org/10.1007/s10878-012-9506-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-012-9506-7