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Daniela Bubboloni

Publications and source records attributed to Daniela Bubboloni.

27 records · Page 2Linked to original sources
arXiv2016

Graph homomorphisms and components of quotient graphs

We study how the number $c(X)$ of components of a graph $X$ can be expressed through the number and properties of the components of a quotient graph $X/\sim.$ We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing $c(X)$ when the projection on the quotient $X/\sim$ is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to $X/\sim$ is an orbit partition.

math.CO↗
arXiv2016

Resolute refinements of social choice correspondences

Many classical social choice correspondences are resolute only in the case of two alternatives and an odd number of individuals. Thus, in most cases, they admit several resolute refinements, each of them naturally interpreted as a tie-breaking rule, satisfying different properties. In this paper we look for classes of social choice correspondences which admit resolute refinements fulfilling suitable versions of anonymity and neutrality. In particular, supposing that individuals and alternatives have been exogenously partitioned into subcommittees and subclasses, we find out arithmetical conditions on the sizes of subcommittees and subclasses that are necessary and sufficient for making any social choice correspondence which is efficient, anonymous with respect to subcommittees, neutral with respect to subclasses and possibly immune to the reversal bias admit a resolute refinement sharing the same properties.

econ.GN↗
arXiv2015

On the reversal bias of the Minimax social choice correspondence

We introduce three different qualifications of the reversal bias in the framework of social choice correspondences. For each of them, we prove that the Minimax social choice correspondence is immune to it if and only if the number of voters and the number of alternatives satisfy suitable arithmetical conditions. We prove those facts thanks to a new characterization of the Minimax social choice correspondence and using a graph theory approach. We discuss the same issue for the Borda and Copeland social choice correspondences.

math.CO↗
arXiv2015

Symmetric majority rules

In the standard arrovian framework and under the assumption that individual preferences and social outcomes are linear orders on the set of alternatives, we study the rules which satisfy suitable symmetries and obey the majority principle. In particular, supposing that individuals and alternatives are exogenously partitioned into subcommittees and subclasses, we provide necessary and sufficient conditions for the existence of reversal symmetric majority rules that are anonymous and neutral with respect to the considered partitions. We also determine a general method for constructing and counting those rules and we explicitly apply it to some simple cases.

math.GR↗
arXiv2013

Conjectures on the normal covering number of the finite symmetric and alternating groups

Let $γ(S_n)$ be the minimum number of proper subgroups $H_i$ of the symmetric group $S_n$ such that each element in $S_n$ lies in some conjugate of one of the $H_i.$ In this paper we conjecture that $$γ(S_n)=\frac{n}{2}\left(1-\frac{1}{p_1}\right) \left(1-\frac{1}{p_2}\right)+2,$$ where $p_1,p_2$ are the two smallest primes in the factorization of $n$ and $n$ is neither a prime power nor a product of two primes. Support for the conjecture is given by a previous result for $n=p_1^{α_1}p_2^{α_2},$ with $(α_1,α_2)\neq (1,1)$. We give further evidence by confirming the conjecture for integers of the form $n=15q$ for an infinite set of primes $q$, and by reporting on a Magma computation. We make a similar conjecture for $γ(A_n)$, when $n$ is even, and provide a similar amount of evidence.

math.GR↗
arXiv2013

Normal coverings and pairwise generation of finite alternating and symmetric groups

The normal covering number $γ(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$ such that, for $G$ a symmetric group $\Sym(n)$ or an alternating group $\Alt(n)$, $γ(G)\geq cn$. This improves results of the first two authors who had earlier proved that $aφ(n)\leqγ(G)\leq 2n/3,$ for some positive constant $a$, where $φ$ is the Euler totient function. Bounds are also obtained for the maximum size $κ(G)$ of a set $X$ of conjugacy classes of $G=\Sym(n)$ or $\Alt(n)$ such that any pair of elements from distinct classes in $X$ generates $G$, namely $cn\leq κ(G)\leq 2n/3$.

math.GR↗
arXiv2012

Compositions of n Satisfying Some Coprimality Conditions

A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic formula for the number of k-compositions having the first summand coprime to the others. Then, we estimate the number of k-compositions whose summands are all pairwise coprime.

math.NT↗
arXiv2010

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups H_i for i=1 ...k of the symmetric group S_n (or the alternating group A_n) such that each element in the group S_n (respectively A_n) lies in some conjugate of one of the H_i. We prove that this number lies between a.phi(n) and bn for certain constants a, b, where phi(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+phi(n)/2, and we determine the exact value for S_n when n is odd and for A_n when n is even.

math.GR↗