SIAM Journal on Discrete Mathematics

We discuss several parametrizations of the space of circular planar electrical networks. With any circular planar network we associate a canonical minimal network with the same response matrix, called a “standard” network. The conductances of edges in a ...

We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every $\varepsilon>0$, with high probability (w.h.p.) $O(\log n)$ rounds are ...

There are a variety of existing conditions for a degree sequence to be graphic. When a degree sequence satisfies any of these conditions, there exists a graph that realizes the sequence. We formulate several novel sufficient graphicality criteria that ...

Finding inclusion-minimal hitting sets (MHSs) for a given family of sets is a fundamental combinatorial problem with applications in domains as diverse as Boolean algebra, computational biology, and data mining. Although many algorithms are available in ...

For scheduling problems on parallel machines, the power of preemption is defined as the supremum ratio of the cost of an optimal nonpreemptive schedule over the cost of an optimal preemptive schedule (for the same input), where the cost is defined by a fixed ...

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Turán number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order of a graph is ...

Green [B. Green, Combinatorica, 25 (2005), pp. 307--326] showed that there exist constants $C_1,C_2>0$ such that the clique number $\omega_n$ of the Cayley graph on $\mathbb{F}_2^n$ generated by a random subset satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\...

We show that for any fixed dense graph $G$ and bounded-degree tree $T$ on the same number of vertices, a modest random perturbation of $G$ will typically contain a copy of $T$. This combines the viewpoints of the well-studied problems of embedding trees into ...

The dimension of a partial order $P$ is the minimum number of linear orders whose intersection is $P$. There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [SIAM J. Algebraic Discrete Methods, 3 (1982), pp. ...

A colored eulerian digraph is an eulerian digraph $G$ where a color is assigned to the tail of each edge and a color is assigned to the head of each edge. A compatible circuit is an eulerian circuit such that for every two consecutive edges $uv$ and $vw$ ...

Let $G$ be an interval graph and take one of its vertices $x$. Can we find in linear time a minimum number of vertex disjoint paths of $G$ which cover the vertex set of $G$ and have $x$ as one of their endpoints? This paper provides a positive answer to ...

The paper is devoted to the normal tiling whose tiles are uniformly bounded and general connected closed sets instead of being restricted to polytopes or convex sets. We estimate the number of neighbors of a tile in the normal tiling and develop various ...

A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial templates that are ...

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched ...

We introduce a new technique to study pattern avoidance in dynamical systems, namely, the use of a commuter function between nonconjugate dynamical systems. We investigate the properties of such a commuter function, specifically $h : [0,1] \to [0,1]$ ...

We are given a symmetric crossing supermodular set function $p$ on $V$ and a partition $\mathcal{P}$ of $V$. We solve the problem of finding a graph with vertex set $V$ having edges only between the classes of $\mathcal{P}$ such that for every subset $X$ ...

A famous conjecture of Pósa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy, and Szemerédi [Random ...

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $\|r_...

Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a cycle of $G$ with minimum weight. The best previously known algorithm for ...

The well-studied “power of two choices” family of algorithms creates balanced allocations of $m$ balls into $n$ bins by, for each ball, selecting a few bins at random and then placing the item in the least-loaded bin. A natural variation is to create an ...

Let $G_{n,p}$ be the standard Erdös--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 ...

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a nonnegative integer $k,$ decide whether the cyclability of $G$ is at least $k,$ is NP-...

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined ...

Chang's lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of Chang's lemma that yields ...

Sensor networks are ubiquitously used for detection and tracking and, as a result, covering is one of the main tasks of such networks. We study the problem of maximizing the coverage lifetime of a barrier by mobile sensors with limited battery power, where ...

The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al., is an estimate of the multiset of probabilities in an unknown probability distribution $\mathbf{p}$, the estimate being obtained from $n$ independent and identically distributed ...

Given a complete graph $K_{n}=(V, E)$ with nonnegative edge costs $c\in {\mathbb R}^{E}$, the problem 2EC is that of finding a 2-edge connected spanning multisubgraph of $K_{n}$ of minimum cost. The integrality gap $\alpha\text{2{\it EC}}$ of the linear ...

In this note we present a corrected version of Lemma 4.9 and two corollaries implied by this lemma, from our paper [Bondy, Lonc, and Rzaͅżewski, SIAM J. Discrete Math., 30 (2016), pp. 452--464].