arXiv subjects

Alan Frieze

Publications and source records attributed to Alan Frieze.

At least 127 records · Page 7Linked to original sources

arXiv2016

The emergence of a giant component in random subgraphs of pseudo-random graphs

Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the adjacency matrix of $G$ is such that the eigenvalue $λ$ which is second largest in absolute value satisfies $λ=o(d)$. Let $G_p$ with $p=\fracα{d}$ be obtained from $G$ by including each edge of $G$ independently with probability $p$. We show that if $α<1$ then whp the maximum component size of $G_p$ is $O(\log n)$ and if $α>1$ then $G_p$ contains a unique giant component of size $Ω(n)$, with all other components of size $O(\log n)$.

math.CO↗

arXiv2016

How many random edges make a dense graph hamiltonian?

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $Θ(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.

math.CO↗

arXiv2016

Adding random edges to dense graphs

This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and $k$-connectivity.

math.CO↗

arXiv2016

Looking for vertex number one

Given an instance of the preferential attachment graph $G_n=([n],E_n)$, we would like to find vertex 1, using only 'local' information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al gave an an algorithm which runs in time $O(\log^4 n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^4 n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(ω\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $ω=ω(n)$ is any function that goes to infinity with $n$.

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

Minimum-cost matching in a random graph with random costs

Let $G_{n,p}$ be the standard Erdős-Rényi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$, suppose that each edge $e\in E$ is given an independent uniform exponential rate one cost. Let $C(G)$ denote the random variable equal to the length of the minimum cost perfect matching, assuming that $G$ contains at least one. We show that w.h.p. if $d=np\gg(\log n)^2$ then w.h.p. ${\bf E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}$. This generalises the well-known result for the case $G=K_{n,n}$. We also show that w.h.p. ${\bf E}[C(G_{n,p})] =(1+o(1))\frac{\p^2}{12p}$ along with concentration results for both types of random graph.

math.CO↗

arXiv2015

Impartial Redistricting: A Markov Chain Approach

The gerrymandering problem is a worldwide problem which sets great threat to democracy and justice in district based elections. Thanks to partisan redistricting commissions, district boundaries are often manipulated to benefit incumbents. Since an independent commission is hard to come by, the possibility of impartially generating districts with a computer is explored in this thesis. We have developed an algorithm to randomly produce legal redistricting schemes for Pennsylvania.

cs.CY↗

arXiv2015

Vacant sets and vacant nets: Component structures induced by a random walk

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$.%These sets induce subgraphs of the underlying graph. Let $Γ(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat Γ(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk, the graph $Γ(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk, such that for $t\leq (1-ε)t^*$, $Γ(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+ε)t^*$ all the components of $Γ(t)$ are of size $O(\log n)$. We establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat Γ(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and non-backtracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.

math.CO↗

arXiv2015

Separating subadditive Euclidean functionals

If we are given $n$ random points in the hypercube $[0,1]^d$, then the minimum length of a Traveling Salesperson Tour through the points, the minimum length of a spanning tree, and the minimum length of a matching, etc., are known to be asymptotically $βn^{\frac{d-1}{d}}$ a.s., where $β$ is an absolute constant in each case. We prove separation results for these constants. In particular, concerning the constants $β_{\mathrm{TSP}}^d$, $β_{\mathrm{MST}}^d$, $β_{\mathrm{MM}}^d$, and $β_{\mathrm{TF}}^d$ from the asymptotic formulas for the minimum length TSP, spanning tree, matching, and 2-factor, respectively, we prove that $β_{\mathrm{MST}}^d<β_{\mathrm{TSP}}^d$, $2β_{\mathrm{MM}}^d<β_{\mathrm{TSP}}^d$, and $β_{\mathrm{TF}}^d<β_{\mathrm{TSP}}^d$ for all $d\geq 2$. We also asymptotically separate the TSP from its linear programming relaxation in this setting. Our results have some computational relevance, showing that a certain natural class of simple algorithms cannot solve the random Euclidean TSP efficiently.

math.PR↗

arXiv2015

On-line list colouring of random graphs

In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.

math.CO↗

arXiv2015

On the chromatic number of a random hypergraph

We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.

cs.DM↗

arXiv2014

Rainbow Connection of Random Regular Graphs

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove that with probability tending to one as $n$ goes to infinity the rainbow connection of $G$ satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.

math.CO↗

arXiv2014

Traveling in randomly embedded random graphs

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood-Halton-Hammersley theorem to this setting.

math.PR↗

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

A note on the vacant set of random walks on the hypercube and other regular graphs of high degree

We consider a random walk on a $d$-regular graph $G$ where $d\to\infty$ and $G$ satisfies certain conditions. Our prime example is the $d$-dimensional hypercube, which has $n=2^d$ vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let $Λ(t)$ be the subgraph induced by the vacant set of the walk at step $t$. We show that if certain conditions are satisfied then the graph $Λ(t)$ undergoes a phase transition at around $t^*=n\log_ed$. Our results are that if $t\leq(1-ε)t^*$ then w.h.p. as the number vertices $n\to\infty$, the size $L_1(t)$ of the largest component satisfies $L_1\gg\log n$ whereas if $t\geq(1+\e)t^*$ then $L_1(t)=o(\log n)$.

math.CO↗

arXiv2014

The height of random $k$-trees and related branching processes

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to clog t, where t is the number of vertices, and c=c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k-->oo the height of both processes is asymptotic to log t/(k log 2).

math.CO↗

arXiv2014

Cover time of a random graph with a degree sequence II: Allowing vertices of degree two

We study the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\mathbf{d}=(d_i)_{i=1}^n$. In a previous work, the asymptotic cover time was obtained under a number of assumptions on $\mathbf{d}$, the most significant being that $d_i\geq 3$ for all $i$. Here we replace this assumption by $d_i\geq 2$. As a corollary, we establish the asymptotic cover time for the 2-core of the emerging giant component of $\mathcal{G}(n,p)$.

math.CO↗