scholarly journals On minimum integer representations of weighted games

2014 ◽  
Vol 67 ◽  
pp. 9-22 ◽  
Author(s):  
Josep Freixas ◽  
Sascha Kurz
Keyword(s):  
Minimum Integer ◽  
Weighted Games

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

scholarly journals Simple games and weighted games: A theoretical and computational viewpoint

2009 ◽  
Vol 157 (7) ◽  
pp. 1496-1508 ◽  
Author(s):  
Josep Freixas ◽  
Xavier Molinero
Keyword(s):  
Simple Games ◽  
Weighted Games

Probability and Bias in Generating Supersoluble Groups

2015 ◽  
Vol 59 (4) ◽  
pp. 899-909 ◽  
Author(s):  
Eleonora Crestani ◽  
Giovanni De Franceschi ◽  
Andrea Lucchini
Keyword(s):  
Uniform Distribution ◽  
Supersoluble Group ◽  
Supersoluble Groups ◽  
Generating Set ◽  
Minimum Integer

AbstractWe discuss some questions related to the generation of supersoluble groups. First we prove that the number of elements needed to generate a finite supersoluble groupGwith good probability can be quite a lot larger than the smallest cardinality d(G) of a generating set ofG. Indeed, ifGis the free prosupersoluble group of rankd⩾ 2 and dP(G) is the minimum integerksuch that the probability of generatingGwithkelements is positive, then dP(G) = 2d+ 1. In contrast to this, ifk–d(G) ⩾ 3, then the distribution of the first component in ak-tuple chosen uniformly in the set of all thek-tuples generatingGis not too far from the uniform distribution.

Choice Numbers of Graphs: a Probabilistic Approach

1992 ◽  
Vol 1 (2) ◽  
pp. 107-114 ◽  
Author(s):  
Noga Alon
Keyword(s):  
Probabilistic Approach ◽  
Probabilistic Methods ◽  
Proper Coloring ◽  
Vertex Graph ◽  
Minimum Integer ◽  
Choice Number ◽  
Almost All

The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

SOME OPEN PROBLEMS IN SIMPLE GAMES

2013 ◽  
Vol 15 (02) ◽  
pp. 1340005 ◽  
Author(s):  
CESARINO BERTINI ◽  
JOSEP FREIXAS ◽  
GIANFRANCO GAMBARELLI ◽  
IZABELLA STACH
Keyword(s):  
Power Indices ◽  
Review Of Literature ◽  
Simple Games ◽  
Open Problems ◽  
Weighted Games

This paper presents a review of literature on simple games and highlights various open problems concerning such games; in particular, weighted games and power indices.

On the Complexity of the Decisive Problem in Simple and Weighted Games

2011 ◽  
Vol 37 ◽  
pp. 21-26 ◽  
Author(s):  
Fabián Riquelme ◽  
Andreas Polyméris
Keyword(s):  
Weighted Games
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