Abstract
A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order \(n\ge 9\) and size at least \(\lfloor \frac{7n-13}{2}\rfloor \) contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order \(n\ge 6\) and size at least \(3n-5\) contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both is sharp in general.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bialostocki, A., Finkel, D., Gyárfás, A.: Disjoint chorded cycles in graphs. Dicrete Math. 308, 5886–5890 (2008)
Bondy, J.A., Murty, U.S.R.: Graph Theory, 2nd edn. Springer, New York (2008)
Brown, J.I., Hickman, C., Sokal, A.D., Wagner, D.G.: On the chromatic roots of gneralized theta graphs. J. Combin. Theory, Ser. B 83, 272–297 (2001)
Chiba, S., Fujita, S., Gao, Y., Li, G.: On a sharp degree sum condition for disjoint chorded cycles in graphs. Graphs Combin. 26, 173–186 (2010)
Eichhorn, D., Mubayi, D., O’Bryant, K., West, D.B.: The edge-bandwidth of theta graphs. J. Graph Theory 35, 89–98 (2000)
Erdős, P.: Extremal problems in graph theory. In: Harary, F. (ed.) A seminar in graph theory. Holt, Rinehart and winston, New York (1967)
Erdős, P.: On sequences of integers no one of which divides the product of two others and on some related problems. Inst. Math. Mech. Univ. Tomsk 2, 74–83 (1938)
Fujita, S., Magnant, C.: Independence number and disjoint theta graphs. Electron. J. Combin. 18(150), 1 (2011)
Kawarabayashi, K.: \(K_{4}^{-}\)-factor in graphs. J. Graph Theory 39, 111–128 (2002)
Peck, G.W., Shastri, A.: Bandwidth of theta graphs with shor paths. Discrete Math. 103, 177–187 (1992)
Pósa, L.: On the circuits of finite graphs, Magyar Tud. Akad. Mat. Kut. Int. Közl. 8, 355–361 (1964)
Verstraëte, J.: On arithmetic progessions of cycle lengths in graphs. Combin. Probab. Comput. 9, 369–373 (2000)
Zou, Q.S., Chen, H.Y., Li, G.: Vertex-disjoint cycles of order eight with chords in a bipartite graph. Bull. Malays. Math. Sci. Soc. 36(2), 255–262 (2013)
Acknowledgments
The authors would like to thank the referees for their detailed corrections and helpful suggestions. Supported by National Natural Science Foundation of China (Grant No. 11161035), Ningxia Ziran (Grant No. NZ1153) and research grant from Ningxia University under Grant number: ndzr10-19.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Xueliang Li.
Rights and permissions
About this article
Cite this article
Gao, Y., Ji, N. The Extremal Function for Two Disjoint Cycles. Bull. Malays. Math. Sci. Soc. 38, 1425–1438 (2015). https://doi.org/10.1007/s40840-014-0102-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-014-0102-0