arXiv subjects

Noga Alon

Publications and source records attributed to Noga Alon.

194 records · Page 11Linked to original sources

arXiv2007

On graphs with subgraphs of large independence numbers

Let G be a graph on n vertices in which every induced subgraph on s=\log^3 n vertices has an independent set of size at least t=\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=Θ(\log^2 n/\log \log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.

math.CO↗

arXiv2007

Embedding nearly-spanning bounded degree trees

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<ε<1, there exists a constant c=c(d,ε) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-ε)n vertices with maximum degree at most d. We also prove that if an (n,D,λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λin their absolute values) has large enough spectral gap D/λas a function of d and ε, then G has a copy of every tree T as above.

math.CO↗

arXiv2007

Poisson approximation for non-backtracking random walks

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. Alon, Benjamini, Lubetzky and Sodin studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length $n$ on a high-girth $n$-vertex regular expander, is typically $(1+o(1))\frac{\log n}{\log\log n}$, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in Alon et al. Let $N_t$ denote the number of vertices visited precisely $t$ times by a non-backtracking random walk of length $n$ on a regular $n$-vertex expander of fixed degree and girth $g$. We prove that if $g=ω(1)$, then for any fixed $t$, $N_t/n$ is typically $\frac{1}{\mathrm{e}t!}+o(1)$. Furthermore, if $g=Ω(\log\log n)$, then $N_t/n$ is typically $\frac{1+o(1)}{\mathrm{e}t!}$ uniformly on all $t \leq (1-o(1))\frac{\log n}{\log\log n}$ and 0 for all $t \geq (1+o(1))\frac{\log n}{\log\log n}$. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.

math.PR↗

arXiv2007

Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves.

cs.DS↗

arXiv2006

Non-backtracking random walks mix faster

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if $G$ is a high-girth regular expander on $n$ vertices, then a typical non-backtracking random walk of length $n$ on $G$ does not visit a vertex more than $(1+o(1))\frac{\log n}{\log\log n}$ times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing $n$ balls to $n$ bins uniformly, in contrast to the simple random walk on $G$, which almost surely visits some vertex $Ω(\log n)$ times.

math.PR↗

arXiv2006

Uniformly cross intersecting families

Let $\mathcal{A}$ and $\matchcal{B}$ denote two families of subsets of an $n$-element set. The pair $(\mathcal{A},\mathcal{B})$ is said to be $\ell$-cross-intersecting iff $|A\cap B| = \ell$ for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$. Denote by $P_\ell(n)$ the maximum value of $|\mathcal{A}||\mathcal{B}|$ over all such pairs. The best known upper bound on $P_\ell(n)$ is $Θ(2^n)$, by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all $n \geq 2\ell$, a simple construction of an $\ell$-cross-intersecting pair $(\mathcal{A},\mathcal{B})$ with $|\mathcal{A}||\mathcal{B}| = \binom{2\ell}{\ell}2^{n-2\ell}=Θ(2^n/\sqrt{\ell})$, and conjectured that this is best possible. Consequently, Sgall asked whether or not $P_\ell(n)$ decreases with $\ell$. In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large $\ell$, implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of $\mathcal{A},\mathcal{B}$ over $\mathbb{R}$, we show that there exists some $\ell_0>0$, such that $P_\ell(n) \leq \binom{2\ell}{\ell}2^{n-2\ell}$ for all $\ell \geq \ell_0$. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

math.CO↗

arXiv2006

Privileged users in zero-error transmission over a noisy channel

The $k$-th power of a graph $G$ is the graph whose vertex set is $V(G)^k$, where two distinct $k$-tuples are adjacent iff they are equal or adjacent in $G$ in each coordinate. The Shannon capacity of $G$, $c(G)$, is $\lim_{k\to\infty}α(G^k)^{1/k}$, where $α(G)$ denotes the independence number of $G$. When $G$ is the characteristic graph of a channel $\mathcal{C}$, $c(G)$ measures the effective alphabet size of $\mathcal{C}$ in a zero-error protocol. A sum of channels, $\mathcal{C}=\sum_i \mathcal{C}_i$, describes a setting when there are $t\geq 2$ senders, each with his own channel $\mathcal{C}_i$, and each letter in a word can be selected from either of the channels. This corresponds to a disjoint union of the characteristic graphs, $G=\sum_i G_i$. We show that for any fixed $t$ and any family $F$ of subsets of $T={1,2,...,t}$, there are $t$ graphs $G_1,G_2, ...,G_t$, so that for every subset $I$ of $T$, the Shannon capacity of the disjoint union $\sum_{i \in I} G_i$ is "large" if $I$ contains a member of $F$, and is "small" otherwise.

math.CO↗

arXiv2006

Independent sets in tensor graph powers

The tensor product of two graphs, $G$ and $H$, has a vertex set $V(G)\times V(H)$ and an edge between $(u,v)$ and $(u',v')$ iff both $u u' \in E(G)$ and $v v' \in E(H)$. Let $A(G)$ denote the limit of the independence ratios of tensor powers of $G$, $\lim α(G^n)/|V(G^n)|$. This parameter was introduced by Brown, Nowakowski and Rall, who showed that $A(G)$ is lower bounded by the vertex expansion ratio of independent sets of $G$. In this note we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products.

math.CO↗

arXiv2006

The Shannon capacity of a graph and the independence numbers of its powers

The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed graph can exhibit a complex structure, implying that the Shannon Capacity of a graph cannot be approximated (up to a sub-polynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the series. This is true even if this prefix shows a significant increase of the independence number at a given power, after which it stabilizes for a while.

cs.IT↗

arXiv2006

Graph powers, Delsarte, Hoffman, Ramsey and Shannon

The $k$-th $p$-power of a graph $G$ is the graph on the vertex set $V(G)^k$, where two $k$-tuples are adjacent iff the number of their coordinates which are adjacent in $G$ is not congruent to 0 modulo $p$. The clique number of powers of $G$ is poly-logarithmic in the number of vertices, thus graphs with small independence numbers in their $p$-powers do not contain large homogenous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of Alon and Lubetzky up to a factor of 2. For precise bounds on some graphs, we apply Delsarte's linear programming bound and Hoffman's eigenvalue bound. Finally, we show that for any nontrivial graph $G$, one can point out specific induced subgraphs of large $p$-powers of $G$ with neither a large clique nor a large independent set. We prove that the larger the Shannon capacity of $\bar{G}$ is, the larger these subgraphs are, and if $G$ is the complete graph, then some $p$-power of $G$ matches the bounds of the Frankl-Wilson Ramsey construction, and is in fact a subgraph of a variant of that construction.

math.CO↗

arXiv2005

Percolation on finite graphs and isoperimetric inequalities

Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|G_n|, with probability going to one, uniformly in p. The method from Ajtai, Komlos and Szemeredi [Combinatorica 2 (1982) 1-7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.

math.PR↗

arXiv2002

Discrete mathematics: methods and challenges

Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of modern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas.

math.CO↗

arXiv1998

Packing Ferrers Shapes

Answering a question of Wilf, we show that if $n$ is sufficiently large, then one cannot cover an $n \times p(n)$ rectangle using each of the $p(n)$ distinct Ferrers shapes of size $n$ exactly once. Moreover, the maximum number of pairwise distinct, non-overlapping Ferrers shapes that can be packed in such a rectangle is only $Θ(p(n)/ \log n).$

math.CO↗

arXiv1992

Piercing convex sets

A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. It is shown that for every $p\ge q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family $\scr F$ of compact, convex sets in $R^d$ that has the $(p,q)$ property there is a set of at most $c$ points in $R^d$ that intersects each member of $\scr F$. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.

math.MG↗