Opuscula Math. 35, no. 6 (2015), 907-914
http://dx.doi.org/10.7494/OpMath.2015.35.6.907

Opuscula Mathematica

On vertex stability of complete k-partite graphs

Mateusz Nikodem

Abstract. Let $$H$$ be any graph. We say that graph $$G$$ is $$H$$-stable if $$G-u$$ contains a subgraph isomorphic to $$H$$ for an arbitrary chosen $$u\in V(G)$$. We characterize all $$H$$-stable graphs of minimal size where $$H$$ is any complete $$k$$-partite graph. Thus, we generalize the results of Dudek and Żak regarding complete bipartite graphs.

Keywords: vertex stability, minimal stable graphs, complete $$k$$-partite graphs.

Mathematics Subject Classification: 05C35, 05C60.

Full text (pdf)

1. S. Cichacz, A. Görlich, M. Nikodem, A. Żak, Lower bound on the size of $$(H;1)$$-vertex stable graphs, Discrete Math 312 (2012) 20, 3026-3029.
2. S. Cichacz, A. Görlich, M. Zwonek, A. Żak, On $$(C_n,k)$$ stable graphs, Electron. J. Comb. 18 (2011) 1, #P205.
3. R. Diestel, Graph Theory, 2nd ed., Springer-Verlag, 2000.
4. A. Dudek, A. Szymański, M. Zwonek, $$(H,k)$$ stable graphs with minimum size, Discuss. Math. Graph Theory 28 (2008), 137-149.
5. A. Dudek, M. Zwonek, $$(H,k)$$ stable bipartite graphs with minimum size, Discuss. Math. Graph Theory 29 (2009), 573-581.
6. A. Dudek, A. Żak, On vertex stability with regard to complete bipartite subgraphs, Discuss. Math. Graph Theory 30 (2010), 663-669.
7. J.-L. Fouquet, H. Thuillier, J-M. Vanherpe, A.P. Wojda, On $$(K_q; k)$$ vertex stable graphs with minimum size, Discrete Math. 312 (2012) 14, 2109-2118.
8. J.-L. Fouquet, H. Thuillier, J-M. Vanherpe, A.P. Wojda, On $$(K_q; k)$$ vertex stable graphs with small $$k$$, Electron. J. Comb. 19 (2012) 2, #P50.
9. A. Żak, On $$(K_q;k)$$-stable graphs, J. Graph Theory 74 (2013) 2, 216-221.
10. A. Żak, General lower bound on the size of $$(H;k)$$-stable graphs, J. Comb. Optim. 29 (2015), 367-372.
• Mateusz Nikodem
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Communicated by Mirko Horňák.
• Revised: 2015-01-19.
• Accepted: 2015-03-16.
• Published online: 2015-06-06.

Mateusz Nikodem, On vertex stability of complete k-partite graphs, Opuscula Math. 35, no. 6 (2015), 907-914, http://dx.doi.org/10.7494/OpMath.2015.35.6.907

a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.