On Selkow’s Bound on the Independence Number of Graphs

Open access

Abstract

For a graph G with vertex set V (G) and independence number α(G), Selkow [A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365] established the famous lower bound vV(G)1d(v)+1(1+max{d(v)d(v)+1-uN(v)1d(u)+1,0}) on α (G), where N(v) and d(v) = |N(v)| denote the neighborhood and the degree of a vertex vV (G), respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here.

[1] N. Alon and J.H. Spencer, The Probabilistic Method (Wiley, New York, 1992).

[2] Y. Caro, New Results on the Independence Number (Technical Report, Tel-Aviv University, 1979).

[3] S.M. Selkow, A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365. doi:10.1016/0012-365X(93)00102-B

[4] V.K. Wei, A Lower Bound on the Stability Number of a Simple Graph (Technical Memorandum, TM 81 - 11217 - 9, Bell laboratories, 1981).

Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

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Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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