Discussiones Mathematicae Graph Theory 33(1) (2013) 101-115
doi: 10.7151/dmgt.1656

Dedicated to Mietek Borowiecki

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On Minimum (Kq,k) Stable Graphs

J.L. Fouquet, H. Thuillier, J.M. Vanherpe

L.I.F.O., Faculté des Sciences, B.P. 6759
Université d'Orléans, 45067 Orléans Cedex 2, France

A.P. Wojda

Wydział Matematyki Stosowanej Zakład Matematyki Dyskretnej
A.G.H., Al. Mickiewicza 30, 30-059 Kraków, Poland


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 Żak in the paper On (Kq;k)-stable graphs, (/10.1002/jgt.21705) has proved a conjecture of Dudek, Szymański 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, κ(q)) stable graphs of minimum size where κ(q) is the maximum value for which for every nonnegative integer k < κ(q) the only (Kq,k) stable graph of minimum size is Kq+k and by determining the exact value of κ(q).

Keywords: stable graphs

2010 Mathematics Subject Classification: 05C035.


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Received 2 May 2012
Revised 30 October 2012
Accepted 30 October 2012