On Minimum (Kq, K) Stable Graphs
J.L. Fouquet ; H. Thuillier ; J.M. Vanherpe ; A.P. Wojda
Discussiones Mathematicae Graph Theory, Tome 33 (2013), p. 101-115 / Harvested from The Polish Digital Mathematics Library
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A graph G is a (Kq, k) stable graph (q ≥ 3) if it contains a Kq after deleting any subset of k vertices (k ≥ 0). Andrzej ˙ Zak in the paper On (Kq; k)-stable graphs, ( doi:/10.1002/jgt.21705) has proved a conjecture of Dudek, Szyma´nski and Zwonek stating that for sufficiently large k the number of edges of a minimum (Kq, k) stable graph is (2q − 3)(k + 1) and that such a graph is isomorphic to sK2q−2 + tK2q−3 where s and t are integers such that s(q − 1) + t(q − 2) − 1 = k. We have proved (Fouquet et al. On (Kq, k) stable graphs with small k, Elektron. J. Combin. 19 (2012) #P50) that for q ≥ 5 and k ≤ q 2 +1 the graph Kq+k is the unique minimum (Kq, k) stable graph. In the present paper we are interested in the (Kq, k(q)) stable graphs of minimum size where k(q) is the maximum value for which for every nonnegative integer k

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:267619 
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J.L. Fouquet; H. Thuillier; J.M. Vanherpe; A.P. Wojda. On Minimum (Kq, K) Stable Graphs. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 101-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1656/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory, 244 ( Springer, Series Graduate Texts in Mathematics, 2008).

[2] A. Dudek, A. Szymański and M. Zwonek, (H, k) stable graphs with minimum size, Discuss. Math. Graph Theory 28 (2008) 137-149. doi:10.7151/dmgt.1397[Crossref]

[3] J.-L. Fouquet, H. Thuillier, J.-M. Vanherpe and A.P. Wojda, On (Kq, k) stable graphs with small k, Electron. J. Combin. 19 (2012) #P50. | Zbl 1244.05123

[4] J.-L. Fouquet, H. Thuillier, J.-M. Vanherpe and A.P. Wojda, On (Kq, k) vertex stable graphs with minimum size, Discrete Math. 312 (2012) 2109-2118. doi:10.1016/j.disc.2011.04.017[WoS][Crossref] | Zbl 1244.05123

[5] P. Frankl and G.Y. Katona, Extremal k-edge-hamiltonian hypergraphs, Discrete Math. 308 (2008) 1415-1424. doi:10.1016/j.disc.2007.07.074[WoS][Crossref]

[6] G.Y. Katona and I. Horváth, Extremal P4-stable graphs, Discrete Appl. Math. 159 (2011) 1786-1792. doi:10.1016/j.dam.2010.11.016[Crossref] | Zbl 1228.05187

[7] J.J. Sylvester, Question 7382, Mathematical Questions from the Educational Times, (1884) 41:21.

[8] A. Żak, On (Kq; k)-stable graphs, J. Graph Theory, (2012),to appear. doi:/10.1002/jgt.21705