Related papers: Thresholds for Extreme Orientability
We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending…
Suppose the edges of the complete graph on $n$ vertices are assigned a uniformly chosen random ordering. Let $X$ denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by…
The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There,we are given a table with $n$ locations, and we…
In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f…
Capacitated Vertex Cover is the hard-capacitated variant of Vertex Cover: given a graph, a capacity for every vertex, and an integer $k$, the task is to select at most $k$ vertices that cover all edges and assign each edge to one of its…
Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is…
A maximum priority matching is a matching in an undirected graph that maximizes a priority score defined with respect to given vertex priorities. An earlier paper showed how to find maximum priority matchings in unweighted graphs. This…
We consider random systems of equations x_1 + ... + x_k = a; 0 <= a <= 2 which are interpreted as equations modulo 3: We show for k >= 15 that the satisfiability threshold of such systems occurs where the 2-core has density 1: We show a…
In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the…
We give an algorithm that decides whether the bipartite crossing number of a given graph is at most $k$. The running time of the algorithm is upper bounded by $2^{O(k)} + n^{O(1)}$, where $n$ is the number of vertices of the input graph,…
Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is \NP-hard. One way to allow for fewer crossings in practice are…
Graph density profiles are fundamental objects in extremal combinatorics. Very few profiles are fully known, and all are two-dimensional. We show that even in high dimensions ratios of graph densities and numbers often form the power-sum…
We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…
Threshold selection is a fundamental problem in any threshold-based extreme value analysis. While models are asymptotically motivated, selecting an appropriate threshold for finite samples is difficult and highly subjective through standard…
Let $G = (V, E)$ be a connected graph, and let $T$ in $V$ be a subset of vertices. An orientation of $G$ is called $T$-odd if any vertex $v \in V$ has odd in-degree if and only if it is in $T$. Finding a T -odd orientation of G can be…
Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold…
The celebrated Corr\'{a}di--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemer\'{e}di Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$…
We give essentially tight bounds for, $\nu(d,k)$, the maximum number of distinct neighbourhoods on a set $X$ of $k$ vertices in a graph with twin-width at most~$d$. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet…
Characterizing the risk of operations is a fundamental requirement in robotics, and a crucial ingredient of safe planning. The problem is multifaceted, with multiple definitions arising in the vast recent literature fitting different…
We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained"…