arXiv subjects

Carlos Hoppen

Publications and source records attributed to Carlos Hoppen.

31 records · Page 2Linked to original sources

arXiv2017

Packing arborescences in random digraphs

We study the problem of packing arborescences in the random digraph $\mathcal D(n,p)$, where each possible arc is included uniformly at random with probability $p=p(n)$. Let $λ(\mathcal D(n,p))$ denote the largest integer $λ\geq 0$ such that, for all $0\leq \ell\leq λ$, we have $\sum_{i=0}^{\ell-1} (\ell-i)|\{v: d^{in}(v) = i\}| \leq \ell$. We show that the maximum number of arc-disjoint arborescences in $\mathcal D(n,p)$ is $λ(\mathcal D(n,p))$ a.a.s. We also give tight estimates for $λ(\mathcal D(n,p))$ depending on the range of $p$.

math.CO↗

arXiv2017

Estimating parameters associated with monotone properties

There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity $q_z=q_z(ε)$ of an estimable parameter $z$ is the size of a random sample of a graph $G$ required to ensure that the value of $z(G)$ may be estimated within an error of $ε$ with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}=\mbox{Forb}(\mathcal{F})$, we study the sample complexity of estimating a bounded graph parameter $z_{\mathcal{P}}$ that, for an input graph $G$, counts the number of spanning subgraphs of $G$ that satisfy $\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

math.CO↗

arXiv2017

Local algorithms, regular graphs of large girth, and random regular graphs

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As examples, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum and minimum bisection, maximum $k$-independent sets, minimum $k$-dominating sets and minimum connected and weakly-connected dominating sets in $r$-regular graphs with large girth.

math.CO↗

arXiv2016

The $p$-spectral radius of the Laplacian

The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as $λ^{(p)}(G)=\max \{x^TAx : \|x\|_p=1 \}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix $L$, and define the $p$-spectral radius of the Laplacian as $μ^{(p)}(G)=\max \{x^TLx : \|x\|_p=1 \}$. We show that $μ^{(p)}(G)$ relates to invariants such as maximum degree and size of a maximum cut. We also show properties of $μ^{(p)}(G)$ as a function of $p$, and a upper bound on $\max_{G \colon |V(G)|=n} μ^{(p)}(G)$ in terms of $n=|V|$ for $p\ge 2$, which is attained if $n$ is even.

math.CO↗

arXiv2016

Densities in large permutations and parameter testing

A classical theorem of Erdos, Lovasz and Spencer asserts that the densities of connected subgraphs in large graphs are independent. We prove an analogue of this theorem for permutations and we then apply the methods used in the proof to give an example of a finitely approximable permutation parameter that is not finitely forcible. The latter answers a question posed by two of the authors and Moreira and Sampaio.

cs.DM↗

arXiv2016

Edge-colorings of graphs avoiding complete graphs with a prescribed coloring

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a copy of $F$ such that the edge partition induced by the coloring is isomorphic to the one of $F$. We think of $\widehat{F}$ as the pattern of coloring that should be avoided. The main question is, for a large enough $n$, to find the (extremal) graph $G$ on $n$ vertices which maximizes $c_{r, \widehat{F}}(G)$. This problem generalizes a question of Erd{\H o}s and Rothschild, who originally asked about the number of colorings not containing a monochromatic clique (which is equivalent to the case where $F$ is a clique and the partition $\widehat{F}$ contains a single class). We use Hölder's Inequality together with Zykov's Symmetrization to prove that, for any $r \geq 2$, $k \geq 3$ and any pattern $\widehat{K_k}$ of the clique $K_k$, there exists a complete multipartite graph that is extremal. Furthermore, if the pattern $\widehat{K_k}$ has at least two classes, with the possible exception of two very small patterns (on three or four vertices), every extremal graph must be a complete multipartite graph. In the case that $r=3$ and $\widehat{F}$ is a rainbow triangle (that is, where $F=K_3$ and each part is a singleton), we show that an extremal graph must be an almost complete graph. Still for $r=3$, we extend a result about monochromatic patterns of Alon, Balogh, Keevash and Sudakov to some patterns that use two of the three colors, finding the exact extremal graph. For the later two results, we use the Regularity and Stability Method.

math.CO↗

arXiv2013

Unicyclic Graphs with equal Laplacian Energy

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of order (square root of n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.

math.CO↗

arXiv2012

Limits of permutation sequences

A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.

math.CO↗

arXiv2012

Characterizing Trees with Large Laplacian Energy

We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer $n$, we find a class of cardinality approximately $\sqrt{n}$ whose elements are the $n$-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum $S_k(T)$ of the $k$ largest Laplacian eigenvalues of an $n$-vertex tree $T$ with diameter at least four, where $k \in \{1,...,n\}$.

math.CO↗

arXiv2011

Limits of permutation sequences through permutation regularity

A permutation sequence $(σ_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $τ$, the density of occurrences of $τ$ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An important ingredient in the proofs is a new model of random permutations, which generalizes previous models and is interesting for its own sake.

math.CO↗

arXiv2011

Hypergraphs with many Kneser colorings (Extended Version)

For fixed positive integers $r, k$ and $\ell$ with $1 \leq \ell < r$ and an $r$-uniform hypergraph $H$, let $κ(H, k,\ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same color class intersect in at least $\ell$ elements. Consider the function $\KC(n,r,k,\ell)=\max_{H\in{\mathcal H}_{n}} κ(H, k,\ell) $, where the maximum runs over the family ${\mathcal H}_n$ of all $r$-uniform hypergraphs on $n$ vertices. In this paper, we determine the asymptotic behavior of the function $\KC(n,r,k,\ell)$ for every fixed $r$, $k$ and $\ell$ and describe the extremal hypergraphs. This variant of a problem of Erdős and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdős--Ko--Rado Theorem on intersecting systems of sets [Intersection Theorems for Systems of Finite Sets, Quarterly Journal of Mathematics, Oxford Series, Series 2, {\bf 12} (1961), 313--320].

math.CO↗

arXiv2007

Induced forests in regular graphs with large girth

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

math.CO↗

arXiv2006

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at least d_C(u,v)-e(n). Let f(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1} \log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph.

math.MG↗