Abstract
It has been shown (J. Harant and D. Rautenbach, Domination in bipartite graphs. Discrete Math. 309:113–122, 2009) that the domination number of a graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5, 7, 10 nor 13 is at most \({\frac{3n}{8}}\). They believed that the assumption that the graphs do not contain cycles of length 10 might be replaced by the exclusion of finitely many exceptional graphs. In this paper, we positively answer that if G is a connected graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5 nor 7 and is not one of three exceptional graphs, then the domination number of G is at most \({\frac{3n}{8}}\).
Similar content being viewed by others
References
Harant J., Rautenbach D.: Domination in bipartite graphs. Discrete Math. 309, 113–122 (2009)
Ore, O.: Theory of graphs. Am. Math. Soc. Colloq. Publ. 38, (1962)
McCuaig W., Shepherd B.: Domination in graphs with minimum degree two. J. Graph Theory 13, 749–762 (1989)
Reed B.: Paths, stars and the number three. Comb. Probab. Comput. 5, 267–276 (1996)
Alon N., Spencer J.: The Probabilistic Method. John Wiley and Sons, Inc., NJ (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Xg., Sohn, M.Y. Domination Number of Graphs Without Small Cycles. Graphs and Combinatorics 27, 821–830 (2011). https://doi.org/10.1007/s00373-010-1004-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-010-1004-z