Signed k-independence in graphs
Lutz Volkmann
Open Mathematics, Tome 12 (2014), p. 517-528 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let k ≥ 2 be an integer. A function f: V(G) → −1, 1 defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α sk(G) of G. In this work, we mainly present upper bounds on α sk (G), as for example α sk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality αSkG+αSkG¯⩽n+2k-3 , where n is the order, Δ(G) the maximum degree and G¯ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:269006 
@article{bwmeta1.element.doi-10_2478_s11533-013-0357-y,
     author = {Lutz Volkmann},
     title = {Signed k-independence in graphs},
     journal = {Open Mathematics},
     volume = {12},
     year = {2014},
     pages = {517-528},
     zbl = {1284.05205},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0357-y}
}

Lutz Volkmann. Signed k-independence in graphs. Open Mathematics, Tome 12 (2014) pp. 517-528. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0357-y/

[1] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 208, Marcel Dekker, New York, 1998 | Zbl 0890.05002

[2] Haynes T.W., Hedetniemi S.T., Slater P.J. (Eds.), Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 209, Marcel Dekker, New York, 1998 | Zbl 0890.05002

[3] Henning M.A., Signed 2-independence in graphs, Discrete Math., 2002, 250(1–3), 93–107 http://dx.doi.org/10.1016/S0012-365X(01)00275-8 | Zbl 1003.05076

[4] Shan E., Sohn M.Y., Kang L., Upper bounds on signed 2-independence numbers of graphs, Ars Combin., 2003, 69, 229–239 | Zbl 1073.05559

[5] Turán P., On an extremal problem in graph theory, Mat. Fiz. Lapok, 1941, 48, 436–452 (in Hungarian)

[6] Volkmann L., Bounds on the signed 2-independence number in graphs, Discuss. Math. Graph Theory, 2013, 33(4), 709–715 http://dx.doi.org/10.7151/dmgt.1686 | Zbl 1295.05182

[7] Zelinka B., On signed 2-independence numbers of graphs (manuscript)