arXiv subjects

Liviu Ilinca

Publications and source records attributed to Liviu Ilinca.

4 recordsLinked to original sources

arXiv2012

Asymptotics of the Upper Matching Conjecture

We give upper bounds for the number $Φ_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular, for $d$-regular, $N$-vertex graphs, our bound is best possible up to an error factor of the form $\exp[o_d(1)N]$, where $o_d(1) \rightarrow 0$ as $d \rightarrow \infty$. This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markström. Some further possibilities are also suggested.

math.CO↗

arXiv2012

Counting maximal antichains and independent sets

Answering several questions of Duffus, Frankl and Rödl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.

math.CO↗

arXiv2010

The number of 3-SAT functions

With $G_k(n)$ the number of functions of $n$ boolean variables definable by $k$-SAT formulae, we prove that $G_3(n)$ is asymptotic to $2^{n+\binom{n}{3}}$. This is a strong form of the case $k=3$ of a conjecture of Bollobás, Brightwell and Leader stating that for fixed $k$, $\log_2 G_k(n)\sim \binom{n}{k}$.

math.CO↗

arXiv2010

On the Number of 2-SAT Functions

We give an alternative proof of a conjecture of Bollobás, Brightwell and Leader, first proved by Peter Allen, stating that the number of boolean functions definable by 2-SAT formulae is $(1+o(1))2^{\binom{n+1}{2}}$. One step in the proof determines the asymptotics of the number of "odd-blue-triangle-free" graphs on $n$ vertices.

math.CO↗