Abstract
A graph is said to be \(C_k\)-saturated if it contains no cycles of length \(k\) but does contain such a cycle after the addition of any edge from the complement of the graph. Determining the minimum size of \(C_k\)-saturated graphs is one of the interesting problems on extremal graphs. The exact minimum sizes are known for \(k=3, 4\) and \(5\), but only general bounds are shown for \(k \ge 6\). This paper deals with bounds of the minimum size when \(k=6\). It is shown that the minimum size of a \(C_6\)-saturated graph on \(n\) vertices is at least \( \lceil \frac{7n}{6}\rceil -2\) and at most \( \lfloor \frac{3n-3}{2} \rfloor \). This lower bound improves the best previously known result.
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Acknowledgments
The authors are grateful to Prof. Katsuhiro Ota for his helpful suggestions on this research, and thank Prof. John R. Schmitt for his comments and information about [15]. We are also indebted to the anonymous referees for their detailed and constructive comments. This research is supported in part by the MEXT Grand in Aid for Scientific Research (C) No. 22510135.
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Zhang, M., Luo, S. & Shigeno, M. On the Number of Edges in a Minimum \(C_6\)-Saturated Graph. Graphs and Combinatorics 31, 1085–1106 (2015). https://doi.org/10.1007/s00373-014-1422-4
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DOI: https://doi.org/10.1007/s00373-014-1422-4