arXiv subjects

Michael Anastos

Publications and source records attributed to Michael Anastos.

40 records · Page 3Linked to original sources

arXiv2017

Randomly coloring simple hypergraphs with fewer colors

We study the problem of constructing a (near) uniform random proper $q$-coloring of a simple $k$-uniform hypergraph with $n$ vertices and maximum degree $Δ$. (Proper in that no edge is mono-colored and simple in that two edges have maximum intersection of size one). We show that if $q\geq \max\{C_k\log n,500k^3Δ^{1/(k-1)}\}$ then the Glauber Dynamics will become close to uniform in $O(n\log n)$ time, given a random (improper) start. This improves on the results in Frieze and Melsted [5].

cs.DM↗

arXiv2017

A Ramsey Property of Random Regular and $k$-out Graphs

In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the other hand, there exists a constant $γ> 0$ such that w.h.p., every $r$-coloring of the edges of $\mathcal{G}(n, 2r+1)$ must contain a monochromatic cycle of length at least $γn$. We prove an analogous result for random $k$-out graphs.

math.CO↗

arXiv2017

Connectivity of the k-out Hypercube

In this paper we study the connectivity properties of the random subgraph of the $n$-cube generated by the $k$-out model and denoted by $Q^n(k)$. Let $k$ be an integer, $1\leq k \leq n-1$. We let $Q^n(k)$ be the graph that is generated by independently including for every $v\in V(Q^n)$ a set of $k$ distinct edges chosen uniformly from all the $\binom{n}{k}$ sets of distinct edges that are incident to $v$. We study connectivity the properties of $Q^n(k)$ as $k$ varies. We show that w.h.p. $Q^n(1)$ does not contain a giant component i.e. a component that spans $Ω(2^n)$ vertices. Thereafter we show that such a component emerges when $k=2$. In addition the giant component spans all but $o(2^n)$ vertices and hence it is unique. We then establish the connectivity threshold found at $k_0= \log_2 n -2\log_2\log_2 n $. The threshold is sharp in the sense that $Q^n(\lfloor k_0\rfloor )$ is disconnected but $Q^n(\lceil k_0\rceil+1)$ is connected w.h.p. Furthermore we show that w.h.p. $Q^n(k)$ is $k$-connected for every $k\geq \lceil k_0\rceil+1$.

math.CO↗

arXiv2016

Purchasing a C_4 online

Let $G$ be a graph with edge set $(e_1,e_2,...e_N)$. We independently associate to each edge $e_i$ of $G$ a cost ${x}_i$ that is drawn from a Uniform [0, 1] distribution. Suppose $\mathcal{F}$ is a set of targeted structures that consists of subgraphs of $G$. We would like to buy a subset of $\mathcal{F}$ at small cost, however we do not know a priori the values of the random variables ${x}_1,...,{x}_N$. Instead, we inspect the random variables $x_i$ one at a time. As soon as we inspect the random variable associated with the cost of an edge we have to decide whether we want to buy that edge or reject it for ever. In the present paper we consider the case where $G$ is the complete graph on $n$ vertices and $\mathcal{F}$ is the set of all $C_4$ -cycles on 4 vertices- out of which we want to buy one.

cs.DS↗