# On Minimum (Kq, K) Stable Graphs

J.L. Fouquet; H. Thuillier; J.M. Vanherpe; A.P. Wojda

Discussiones Mathematicae Graph Theory (2013)

- Volume: 33, Issue: 1, page 101-115
- ISSN: 2083-5892

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topJ.L. Fouquet, et al. "On Minimum (Kq, K) Stable Graphs." Discussiones Mathematicae Graph Theory 33.1 (2013): 101-115. <http://eudml.org/doc/267619>.

@article{J2013,

abstract = {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},

author = {J.L. Fouquet, H. Thuillier, J.M. Vanherpe, A.P. Wojda},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {stable graphs},

language = {eng},

number = {1},

pages = {101-115},

title = {On Minimum (Kq, K) Stable Graphs},

url = {http://eudml.org/doc/267619},

volume = {33},

year = {2013},

}

TY - JOUR

AU - J.L. Fouquet

AU - H. Thuillier

AU - J.M. Vanherpe

AU - A.P. Wojda

TI - On Minimum (Kq, K) Stable Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2013

VL - 33

IS - 1

SP - 101

EP - 115

AB - 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

LA - eng

KW - stable graphs

UR - http://eudml.org/doc/267619

ER -

## References

top- [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. Zbl1244.05123
- [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] Zbl1244.05123
- [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] Zbl1228.05187
- [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

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