arXiv subjects

Andrzej Żak

Publications and source records attributed to Andrzej Żak.

7 recordsLinked to original sources

arXiv2026

The Erdős four-edge intersection problem

For an $n$-vertex graph $G$ and a permutation $σ$ of its vertex set, let $σ(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(σ)=|E(G)\cap E(σ(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(σ)\geq k$ for every $σ$. In 1977 Erdős asked whether $f(n,4)=2n-4$, observing that $K_{2,n-2}$ gives the upper bound. We prove that, for all sufficiently large $n$, \[ f(n,4)=2n-4. \] Equivalently, every sufficiently large $n$-vertex graph with at most $2n-5$ edges has a relabelling with at most three common edges. Our proof is inspired by the recent work of Fang and Hou on the Erdős--Mullin five-edge intersection problem and builds on their core--buffer and absorption framework. The main additional ingredients are a growing high-degree core $C$ satisfying \[ |C|Δ(G-C)=o(n), \] and a rigidity analysis of the equality case in the relevant first-moment estimate. This analysis shows that the only core--buffer configuration forcing four local common edges is of $K_{2,|C|}$ type; the strict bound $e(G)\leq2n-5$ then supplies a defect which breaks this configuration.

math.CO↗

arXiv2026

An asymptotic solution to the Erdős four-edge intersection problem

For an $n$-vertex graph $G$ and a permutation $σ$ of its vertex set, let $σ(G)$ denote the corresponding relabelling of $G$, and put $I_G(σ)=|E(G)\cap E(σ(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(σ)\geq k$ for every $σ$. In his 1977 formulation of the problem, Erdős discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erdős's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.

math.CO↗

arXiv2026

Further progress on Wojda's conjecture

Two digraphs of order $n$ are said to pack if they can be found as edge-disjoint subgraphs of the complete digraph of order $n$. It is well established that if the sum of the sizes of the two digraphs is at most $2n-2$, then they pack, with this bound being sharp. However, it is sufficient for the size of the smaller digraph to be only slightly below $n$ for the sum of their sizes to significantly exceed this threshold while still guaranteeing the existence of a packing. In 1985, Wojda conjectured that for any $2 \leq m \leq n/2$, if one digraph has size at most $n - m$ and the other has size less than $2n - \lfloor n/m \rfloor$, then the two digraphs pack. It was previously known that this conjecture holds for $m = Ω(\sqrt{n})$. In this paper, we confirm it for $m \geq 93$ and $n \geq 31m$.

math.CO↗

arXiv2025

Exploring phase transitions and thermal dynamics in nanoconfined liquid crystals using liquid-phase TEM

Nanoconfined liquid crystals (LCs) and their nanocomposites are driving the next generation of photonic applications. Consequently, deepening our understanding of mesophase stability, defect topology, and the dynamic response of LCs at the nanoscale requires the development of novel characterization approaches. This motivates us to perform in situ observations on model 4'-octyl-4-cyanobiphenyl (8CB) LC using liquid-phase scanning transmission electron microscopy (LP-STEM). We find that the electron beam induced consecutive phase changes from smectic A to nematic (SmA-N) and from nematic to isotropic (N-I). The kinetic dependence of the phase transition on dose rate shows that the time between SmA-N and N-I shortens with increasing rate, revealing the hypothesis that a higher electron dose rate increases the energy dissipation rate, leading to substantial heat generation in the sample. We report on the spontaneous formation of disclinations, ordering effects, and complete process reversibility. Radiolytic effects of the electron beam are discussed in detail, and additional experiments with external heating indicate that the observed phenomena are mainly thermal in nature. The results are supported by calculations of heat diffusion, suggesting the nanoconfined 8CB differs significantly in thermal properties compared to the bulk one. This is the first detailed study of LC phase transitions using LP-STEM, which paves the way for further studies of nanoconfined LCs and for the development of the technique for advanced LC materials research.

cond-mat.soft↗

arXiv2023

Constructing sparsest $\ell$-hamiltonian saturated $k$-uniform hypergraphs for a wide range of $\ell$

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.

math.CO↗

arXiv2018

A note on packing of uniform hypergraphs

A packing of two $k$-uniform hypergraphs $H_1$ and $H_2$ is a set $\{H_1', H_2'\}$ of edge-disjoint sub-hypergraphs of the complete $k$-uniform hypergraph $K_n^{(k)}$ such that $H_1'\cong H_1$ and $H_2'\cong H_2$. Whilst the problem of packing of graphs (i.e. 2-uniform hypergraphs) has been studied extensively since seventies with many sharp results, much less is known about packing of general hypergraphs. In this paper we attempt to find the minimum possible sum of sizes $m(n,k)$ of two $k$-uniform, $n$-vertex hypergaphs which do not pack. We also prove a sufficient condition on the product of maximum degrees, which guarantees the packing.

math.CO↗

arXiv2015

Packing large trees of consecutive orders

A conjecture by Bollobás from 1995 (which is a weakenning of the famous Tree Packing Conjecture by Gyárfás from 1976) states that any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $n-i$ vertices, pack into $K_n$, provided $n$ is sufficiently large. We confirm Bollobás conjecture for trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $k-1-i$ leaves or a pending path of order $k-1-i$. As a consequence we obtain that the conjecture is true for $k\leq 5$.

math.CO↗