arXiv subjects

Éva Czabarka

Publications and source records attributed to Éva Czabarka.

26 records · Page 2Linked to original sources

arXiv2016

Paths vs. stars in the local profile of trees

The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p^{(k)}_2(T)$ be the proportion of stars. Our main theorem states: if $p^{(k)}_1(T_n) \to 0$ for a sequence of trees $T_1,T_2,\ldots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to 1$. Both are also shown to be equivalent to the statement that the number of $k$-vertex subtrees grows superlinearly and the statement that the $(k-1)$th degree moment grows superlinearly.

math.CO↗

arXiv2016

Inducibility in binary trees and crossings in random tanglegrams

In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $γ(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to \infty} γ(B,T)$. We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility $1$. We also formulate some open problems and conjectures on the inducibility. Finally, we present an application to crossing numbers of random tanglegrams.

math.CO↗

arXiv2016

On the number of nonisomorphic subtrees of a tree

We show that a tree of order $n$ has at most $O(5^{n/4})$ nonisomorphic subtrees, and that this bound is best possible. We also prove an analogous result for the number of nonisomorphic rooted subtrees of a rooted tree.

math.CO↗

arXiv2015

Reducing Degeneracy in Maximum Entropy Models of Networks

Based on Jaynes' maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry-breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is illustrated on examples.

cond-mat.stat-mech↗

arXiv2015

Abelian groups yield many large families for the diamond problem

There is much recent interest in excluded subposets. Given a fixed poset $P$, how many subsets of $[n]$ can found without a copy of $P$ realized by the subset relation? The hardest and most intensely investigated problem of this kind is when $P$ is a diamond, i.e. the power set of a 2 element set. In this paper, we show infinitely many asymptotically tight constructions using random set families defined from posets based on Abelian groups. They are provided by the convergence of Markov chains on groups. Such constructions suggest that the diamond problem is hard.

math.CO↗

arXiv2013

On Realizations of a Joint Degree Matrix

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is an error in the previous proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix. Finally, we address some of the issues concerning the mixing time of the corresponding MCMC method to sample uniformly from these realizations.

math.CO↗

arXiv2012

Threshold functions for distinct parts: revisiting Erdos-Lehner

We study four problems: put $n$ distinguishable/non-distinguishable balls into $k$ non-empty distinguishable/non-distinguishable boxes randomly. What is the threshold function $k=k(n) $ to make almost sure that no two boxes contain the same number of balls? The non-distinguishable ball problems are very close to the Erd\H os--Lehner asymptotic formula for the number of partitions of the integer $n$ into $k$ parts with $k=o(n^{1/3})$. The problem is motivated by the statistics of an experiment, where we only can tell whether outcomes are identical or different.

math.CO↗

arXiv2012

Mixed orthogonal arrays, $k$-dimensional $M$-part Sperner multi-families, and full multi-transversals

Aydinian et al. [J. Combinatorial Theory A 118(2)(2011), 702-725] substituted the usual BLYM inequality for L-Sperner families with a set of M inequalities for $(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$ type M-part Sperner families and showed that if all inequalities hold with equality, then the family is homogeneous. Aydinian et al. [Australasian J. Comb. 48(2010), 133-141] observed that all inequalities hold with equality if and only if the transversal of the Sperner family corresponds to a simple mixed orthogonal array with constraint M, strength M-1, using $m_i+1$ symbols in the $i^{\text{th}}$ column. In this paper we define $k$-dimensional $M$-part Sperner multi-families with parameters $L_P: P\in\binom{[M]}{k}$ and prove $\binom{M}{k}$ BLYM inequalities for them. We show that if k<M and all inequalities hold with equality, then these multi-families must be homogeneous with profile matrices that are strength M-k mixed orthogonal arrays. For k=M, homogeneity is not always true, but some necessary conditions are given for certain simple families. Following the methods of Aydinian et al. [Australasian J. Comb. 48(2010), 133-141], we give new constructions to simple mixed orthogonal arrays with constraint M, strength M-k, using $m_i+1$ symbols in the ith column. We extend the convex hull method to k-dimensional M-part Sperner multi-families, and allow additional conditions providing new results even for simple 1-part Sperner families.

math.CO↗