arXiv subjects

Rudi Mrazović

Publications and source records attributed to Rudi Mrazović.

12 recordsLinked to original sources

arXiv2026

The inverse problem for the Steiner-Wiener index of trees

For a connected graph $G$ and a set $S\subset V(G)$, the Steiner distance $d_G(S)$ is the minimum number of edges in a connected subgraph of $G$ containing $S$. The Steiner-Wiener $k$ index is defined by $\mathrm{SW}_k(G) = \sum_{S\subset V(G), |S|=k} d_G(S)$. We study the inverse problem for this invariant restricted to trees: for fixed $k$, which positive integers occur as $\mathrm{SW}_k(T)$ for a finite tree $T$? We prove that all sufficiently large positive integers occur as $\mathrm{SW}_k(T)$ for some finite tree $T$ if and only if $k$ is even. For odd $k$, we further show that the set of attainable values has asymptotic density of order $k^{-δ}(\log k)^{-3/2}$, where $δ$ is the Erdős-Tenenbaum-Ford constant.

math.CO↗

arXiv2026

A note on the shortest law for the symmetric group

Let $α(n)$ denote the length of the shortest non-trivial two-variable law for the symmetric group $S_n$. Buskin's quantitative subgroup-separability argument gives the classical lower bound $α(n)\geq 2n-O(1)$. In this short note we give an improvement by proving that $α(n)\geq \frac52 n-O(1)$.

math.CO↗

arXiv2026

Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk

We study the convex hull of planar Brownian motion run until the exit time from the unit disk. Our primary objectives are to compute the expected perimeter and expected area of this convex hull, thereby complementing recent results on the convex hull of reflecting Brownian motion in confined geometries. We reduce the problem of computing the expected perimeter to computing the expected value of the Brownian motion's maximum horizontal displacement at the exit time, and then recast this maximum in terms of harmonic measure in a domain we call the truncated disk. The problem of computing the expected area is reduced to computing the expected value of the difference of squares of the Brownian motion's maximum horizontal displacement at the exit time, and the value of the vertical displacement at the time this maximum horizontal displacement is achieved. In particular, we obtain exact expressions for both the expected perimeter and the expected area. We conclude with further results on the expected areas of two related hulls of the Brownian path run until exiting the disk, namely, the star hull and topological hull.

math.PR↗

arXiv2026

Lines in the prime number graph

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$

math.NT↗

arXiv2025

Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

For a fixed integer n, a pair of nonzero integers {a, c} is called a D(n)-pair if the product ac plus n is a perfect square. In this short note we prove that D(n)-pairs are asymptotically equidistributed (via their associated quadratic forms) among proper SL_2(Z)-equivalence classes of binary quadratic forms of discriminant 4n with fixed content. As a consequence, we obtain a more streamlined and simpler proof of Badesa's asymptotic formula for the number of D(n)-pairs.

math.NT↗

arXiv2023

Transversals in quasirandom latin squares

A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as $n \to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2} + o(1)) n!^2 / n^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

math.CO↗

arXiv2022

An asymptotic for the Hall--Paige conjecture

Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group $G$ of order $n$ satisfying the Hall--Paige condition is $(e^{-1/2} + o(1)) \, |G^\text{ab}| \, n!^2/n^n$.

math.CO↗

arXiv2021

Capacity of the range of random walks on groups

In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem.

math.PR↗

arXiv2016

Additive triples of bijections, or the toroidal semiqueens problem

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $π_1,π_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise sum $π_1+π_2$ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of $\mathbb{Z}/n\mathbb{Z}$, to counting the number of arrangements of $n$ mutually nonattacking semiqueens on an $n\times n$ toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy--Littlewood circle method from analytic number theory, adapted to the group $(\mathbb{Z}/n\mathbb{Z})^n$.

math.CO↗

arXiv2016

A random model for the Paley graph

For a prime $p$ we define the Paley graph to be the graph with the set of vertices $\mathbb{Z}/p\mathbb{Z}$, and with edges connecting vertices whose sum is a quadratic residue. Paley graphs are notoriously difficult to study, particularly finding bounds for their clique numbers. For this reason, it is desirable to have a random model. A well known result of Graham and Ringrose shows that the clique number of the Paley graph is $Ω(\log p\log\log\log p)$ (even $Ω(\log p\log\log p)$, under the generalized Riemann hypothesis) for infinitely many primes $p$ -- a behaviour not detected by the random Cayley graph which is hence deficient as a random model for for the Paley graph. In this paper we give a new probabilistic model which incorporates some multiplicative structure and as a result captures the Graham-Ringrose phenomenon. We prove that if we sample such a random graph independently for every prime, then almost surely (i) for infinitely many primes $p$ the clique number is $Ω(\log p\log\log p)$, whilst (ii) for almost all primes the clique number is $(2+o(1))\log p$.

math.NT↗

arXiv2015

Extractors in Paley graphs: a random model

A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset\mathbb{Z}/p\mathbb{Z}$, each of size at least $\log ^C p$ (for some constant $C$) we have that the number of pairs $(x,y)\in X\times Y$ such that $x+y$ is a quadratic residue modulo $p$ differs from $\frac12|X||Y|$ by $o\left(|X||Y|\right)$. We address the probabilistic analogue of this question, that is for every fixed $δ>0$, given a finite group $G$ and $A\subset G$ a random subset of density $\frac12$, we prove that with high probability for all subsets $|X|,|Y|\geq \log ^{2+δ} |G|$, the number of pairs $(x,y)\in X\times Y$ such that $xy\in A$ differs from $\frac12|X||Y|$ by $o\left(|X||Y|\right)$.

math.CO↗

arXiv2015

One-point concentration of the clique and chromatic numbers of the random Cayley graph on F_2^n

Green showed that there exist constants $C_1,C_2>0$ such that the clique number $ω$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\log n < ω< C_2n\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\mathbb{F}_2^n$ analogue of the famous conjecture by Bollobás and giving almost the complete answer to the question by Green.

math.CO↗