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Patrick Bennett

Publications and source records attributed to Patrick Bennett.

53 records · Page 3Linked to original sources
arXiv2018

A natural barrier in random greedy hypergraph matching

Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most $L$ where $L \ = D/ (\log N)^{ω(1)}$. We consider the random greedy algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\mathcal H}$ that are not saturated by the final matching is at most $ (L/D)^{ \frac{ 1}{ 2(r-1) } + o(1) } $. This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

math.CO↗
arXiv2018

The bipartite $K_{2,2}$-free process and bipartite Ramsey number $b(2, t)$

The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and show that the resulting graph witnesses that $b(2,t) =Ω\left(t^{3/2}/\log t \right)$, thereby improving the best known lower bound.

math.CO↗
arXiv2018

Large monochromatic components and long monochromatic cycles in random hypergraphs

We extend results of Gyárfás and Füredi on the largest monochromatic component in $r$-colored complete $k$-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in $r$-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in $2$-colored complete $k$-uniform hypergraphs.

math.CO↗
arXiv2017

Square of a Hamilton cycle in a random graph

We show that the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle is $p=\frac{1}{\sqrt{n}}$. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić. In addition we consider how many random edges need to be added to a graph of order $n$ with minimum degree $αn$ in order that it contains the square of a Hamilton cycle w.h.p.

math.CO↗
arXiv2016

On the number of alternating paths in bipartite complete graphs

Let $C \subseteq [r]^m$ be a code such that any two words of $C$ have Hamming distance at least $t$. It is not difficult to see that determining a code $C$ with the maximum number of words is equivalent to finding the largest $n$ such that there is an $r$-edge-coloring of $K_{m, n}$ with the property that any pair of vertices in the class of size $n$ has at least $t$ alternating paths (with adjacent edges having different colors) of length $2$. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum $t$ such that there is an $r$-edge-coloring of $K_{m,n}$ such that any pair of vertices in class of size $n$ is connected by $t$ internally disjoint and alternating paths of length $2k$. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs.

math.CO↗
arXiv2016

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow perfect matching is a collection of $n/2$ independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least $1$ (respectively, at least $2$). More precisely, consider $n$ points (i.e. vertices) chosen independently and uniformly at random from the unit $d$-dimensional cube for any fixed $d\ge2$. Form a sequence of graphs on these $n$ vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the $\ell_p$ norm, for any fixed $1<p\le\infty$). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of $\lceil Kn\rceil$ colours, where $K=K(d)$ is a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least $1$ must contain a rainbow perfect matching (for even $n$), and the first graph with minimum degree at least $2$ must contain a rainbow Hamilton cycle.

math.CO↗
arXiv2015

Weak and strong versions of the 1-2-3 conjecture for uniform hypergraphs

Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function $ω:E\to\{1,\dots,w\}$, a coloring of vertices of $H$, induced by $ω$, is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that $H$ is strongly $w$-weighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that $H$ is weakly $w$-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all $5$-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for $r\ge 6$ we show that almost all $r$-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3-uniform hypergraphs are weakly 2-weighted but not 1-weighted and for $r\ge 4$ almost all $r$-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are $r$-uniform hypergraphs which are not strongly $(r^2-r)$-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete.

math.CO↗
arXiv2015

On the Ramsey-Turán number with small $s$-independence number

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Turán number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $α_s(G) < f$, where $α_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^δ)$ for fixed $δ<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<δ< 1$, $RT_s(n,K_{s+1}, n^δ) = Ω(n^{1+δ-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^δ) = O(n^{1+δ})$. Furthermore, the range of $δ$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^δ)$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.

math.CO↗
arXiv2015

The Total Acquisition Number of Random Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ can be moved to a neighbouring vertex $v$, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process. LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = Θ(n)$, confirming a conjecture of West.

math.CO↗
arXiv2015

Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$-CNF $ϕ$ in $n$ variables requires, with high probability, $Ω(n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $ϕ$ requires, with high probability, $Ω(n)$ clauses each of width $Ω(n)$ to be kept at the same time in memory. This gives a $Ω(n^2)$ lower bound for the total space needed in Resolution to refute $ϕ$. These results are best possible (up to a constant factor). The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs $G$ with bipartition $(L,R)$ and left-degree at most 3, $L$ can be covered by certain families of disjoint paths, called VW-matchings, provided that $L$ expands in $R$ by a factor of $(2-ε)$, for $ε< 1/23$.

cs.CC↗
arXiv2014

Rainbow arborescence in random digraphs

We consider the Erdős-Rényi random directed graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let $\mathcal{D}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{D}(n,m)$ independently chooses a colour, taken uniformly at random from a given set of $n(1 + O( \log \log n / \log n)) = n (1+o(1))$ colours. We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{D}(n,M)$ has at most one vertex that has in-degree zero and there are at least $n-1$ distinct colours introduced ($M= n(n-1)$ if at the time when all edges are present there are still less than $n-1$ colours introduced; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{D}(n,M)$ has a rainbow arborescence (that is, a directed, rooted tree on $n$ vertices in which all edges point away from the root and all the edges are different colours). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is "yes".

math.CO↗
arXiv2014

Power of $k$ choices and rainbow spanning trees in random graphs

We consider the Erdős-Rényi random graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let $\mathcal{G}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,..., m$) of $\mathcal{G}(n,m)$ independently chooses precisely $k \in \mathbb{N}$ colours, uniformly at random, from a given set of $n-1$ colours (one may view $e_i$ as a multi-edge). We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{G}(n,M)$ is connected and every colour occurs at least once ($M={n \choose 2}$ if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{G}(n,M)$ has a rainbow spanning tree (that is, multicoloured tree on $n$ vertices). Clearly, both properties are necessary for the desired tree to exist. In 1994, Frieze and McKay investigated the case $k=1$ and the answer to this question is "yes" (asymptotically almost surely). However, since the sharp threshold for connectivity is $\frac {n}{2} \log n$ and the sharp threshold for seeing all the colours is $\frac{n}{k} \log n$, the case $k=2$ is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is "yes" also for $k \ge 2$.

math.CO↗
arXiv2014

Sub-nanosecond signal propagation in anisotropy engineered nanomagnetic logic chains

Energy efficient nanomagnetic logic (NML) computing architectures propagate and process binary information by relying on dipolar field coupling to reorient closely-spaced nanoscale magnets. Signal propagation in nanomagnet chains of various sizes, shapes, and magnetic orientations has been previously characterized by static magnetic imaging experiments with low-speed adiabatic operation; however the mechanisms which determine the final state and their reproducibility over millions of cycles in high-speed operation (sub-ns time scale) have yet to be experimentally investigated. Monitoring NML operation at its ultimate intrinsic speed reveals features undetectable by conventional static imaging including individual nanomagnetic switching events and systematic error nucleation during signal propagation. Here, we present a new study of NML operation in a high speed regime at fast repetition rates. We perform direct imaging of digital signal propagation in permalloy nanomagnet chains with varying degrees of shape-engineered biaxial anisotropy using full-field magnetic soft x-ray transmission microscopy after applying single nanosecond magnetic field pulses. Further, we use time-resolved magnetic photo-emission electron microscopy to evaluate the sub-nanosecond dipolar coupling signal propagation dynamics in optimized chains with 100 ps time resolution as they are cycled with nanosecond field pulses at a rate of 3 MHz. An intrinsic switching time of 100 ps per magnet is observed. These experiments, and accompanying macro-spin and micromagnetic simulations, reveal the underlying physics of NML architectures repetitively operated on nanosecond timescales and identify relevant engineering parameters to optimize performance and reliability.

cond-mat.mes-hall↗
arXiv2014

Speed and Reliability of Nanomagnetic Logic Technology

Nanomagnetic logic is an energy efficient computing architecture that relies on the dipole field coupling of neighboring magnets to transmit and process binary information. In this architecture, nanomagnet chains act as local interconnects. To assess the merits of this technology, the speed and reliability of magnetic signal transmission along these chains must be experimentally determined. In this work, time-resolved pump-probe x-ray photo-emission electron microscopy is used to observe magnetic signal transmission along a chain of nanomagnets. We resolve successive error-free switching events in a single nanomagnet chain at speeds on the order of 100 ps per nanomagnet, consistent with predictions based on micromagnetic modeling. Errors which disrupt transmission are also observed. We discuss the nature of these errors, and approaches for achieving reliable operation.

cond-mat.mes-hall↗
arXiv2012

The t-tone chromatic number of random graphs

A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph $G$, denoted $τ_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\ge Cn^{-1/4}\ln^{9/4}n$, $τ_2(G_{n,p})=(2+o(1))χ(G_{n,p})$ where $χ$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $τ_2(G_{n,p}) = \lceil{{\sqrt{8Δ+1} +5}/{2}}\rceil$ where $Δ$ represents the maximum degree. For the more general concept of $t$-tone coloring, we achieve similar results.

math.CO↗
arXiv2012

A greedy algorithm for finding a large 2-matching on a random cubic graph

A 2-matching of a graph $G$ is a spanning subgraph with maximum degree two. The size of a 2-matching $U$ is the number of edges in $U$ and this is at least $n-\k(U)$ where $n$ is the number of vertices of $G$ and $\k$ denotes the number of components. In this paper, we analyze the performance of a greedy algorithm \textsc{2greedy} for finding a large 2-matching on a random 3-regular graph. We prove that with high probability, the algorithm outputs a 2-matching $U$ with $\k(U) = \tildeΘ\of{n^{1/5}}$.

math.CO↗