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Keywords:
saturation number; saturated graph; linear forest
saturation number; saturated graph; linear forest
Summary:
A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, ${\rm sat} (n, H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs of order $n$. We determine the saturation number ${\rm sat} (n, P_6 + tP_2)$ for $n \geq \frac {10}{3} t + 10$ and characterize the extremal graphs for $n > \frac {10}{3} t + 20$.
References:
[1] Berge, C.: Sur le couplage maximum d'un graphe. C. R. Acad. Sci., Paris 247 (1958), 258-259 French. MR 0100850 | Zbl 0086.16301
[2] Bohman, T., Fonoberova, M., Pikhurko, O.: The saturation function of complete partite graphs. J. Comb. 1 (2010), 149-170. DOI 10.4310/JOC.2010.v1.n2.a5 | MR 2732512 | Zbl 1221.05208
[3] Bollobás, B.: On a conjecture of Erdős, Hajnal and Moon. Am. Math. Mon. 74 (1967), 178-179. DOI 10.2307/2315614 | MR 0205871 | Zbl 0144.45304
[4] Bondy, J. A., Murty, U. S. R.: Graph Theory. Graduate Texts in Mathematics 244. Springer, Berlin (2008). DOI 10.1007/978-1-84628-970-5 | MR 2368647 | Zbl 1134.05001
[5] Chen, G., Faudree, J. R., Faudree, R. J., Gould, R. J., Jacobson, M. S., Magnant, C.: Results and problems on saturation numbers for linear forests. Bull. Inst. Comb. Appl. 75 (2015), 29-46. MR 3444611 | Zbl 1401.05155
[6] Chen, G., Faudree, R. J., Gould, R. J.: Saturation numbers of books. Electron. J. Comb. 15 (2008), Article ID 118, 12 pages. DOI 10.37236/842 | MR 2443133 | Zbl 1158.05033
[7] Chen, Y.-C.: All minimum $C_5$-saturated graphs. J. Graph Theory 67 (2011), 9-26. DOI 10.1002/jgt.20508 | MR 2809558 | Zbl 1223.05130
[8] Chen, Y.-C.: Minimum $K_{2,3}$-saturated graphs. J. Graph Theory 76 (2014), 309-322. DOI 10.1002/jgt.21767 | MR 3218275 | Zbl 1296.05097
[9] Erdős, P., Hajnal, A., Moon, J. W.: A problem in graph theory. Am. Math. Mon. 71 (1964), 1107-1110. DOI 10.2307/2311408 | MR 0170339 | Zbl 0126.39401
[10] Fan, Q., Wang, C.: Saturation numbers for linear forests $P_5\cup tP_2$. Graphs Comb. 31 (2015), 2193-2200. DOI 10.1007/s00373-014-1514-1 | MR 3417226 | Zbl 1328.05099
[11] Faudree, J. R., Faudree, R. J., Gould, R. J., Jacobson, M. S.: Saturation numbers for trees. Electron. J. Comb. 16 (2009), Article ID R91, 19 pages. DOI 10.37236/180 | MR 2529800 | Zbl 1186.05072
[12] Faudree, J. R., Faudree, R. J., Schmitt, J. R.: A survey of minimum saturated graphs. Electron. J. Comb. DS19 (2011), 36 pages. MR 4336221 | Zbl 1222.05113
[13] Faudree, J. R., Ferrara, M., Gould, R. J., Jacobson, M. S.: $tK_p$-saturated graphs of minimum size. Discrete Math. 309 (2009), 5870-5876. DOI 10.1016/j.disc.2008.06.036 | MR 2551965 | Zbl 1229.05141
[14] Kászonyi, L., Tuza, Z.: Saturated graphs with minimal number of edges. J. Graph Theory 10 (1986), 203-210. DOI 10.1002/jgt.3190100209 | MR 0890226 | Zbl 0593.05041
[15] Sullivan, E., Wenger, P. S.: Saturation numbers in tripartite graphs. J. Graph Theory 84 (2017), 428-442. DOI 10.1002/jgt.22033 | MR 3623386 | Zbl 1359.05055
[16] Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 (1941), 436-452 Hungarian. MR 0018405 | Zbl 0026.26903
[17] Tuza, Z.: $C_4$-saturated graphs of minimum size. Acta Univ. Carol., Math. Phys. 30 (1989), 161-167. MR 1046463 | Zbl 0719.05040
[18] West, D. B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (1996). MR 1367739 | Zbl 0845.05001
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