Discussiones Mathematicae Graph Theory 33(4) (2013) 649-656
doi: 10.7151/dmgt.1683

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Some Sharp Bounds on the Negative Decision Number of Graphs

Hongyu Liang

Institute for Interdisciplinary Information Sciences
Tsinghua University
Beijing, China, 100084


Let G = (V,E) be a graph. A function f:V → { −1,1} is called a  of G if ∑u ∈ NG(v)f(u) ≤ 1 for all v ∈ V, where NG(v) denotes the set of neighbors of v in G. The negative decision number of G, introduced in [12], is the maximum value of ∑v ∈ Vf(v) taken over all s of G. In this paper, we present sharp upper bounds on the negative decision number of a graph in terms of its order, minimum degree, and maximum degree. We also establish a sharp Nordhaus-Gaddum-type inequality for the negative decision number.

Keywords: negative decision number, bad function, sharp upper bounds, Nordhaus-Gaddum results

2010 Mathematics Subject Classification: 05C69, 05C75.


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Received 27 July 2011
Revised 20 July 2012
Accepted 23 July 2012