Related papers: k variables are needed to define k-Clique in first…

The $k$-center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly…

We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated $(p,q)$-theorem in discrete geometry. In particular, for a graph $G$ with bounded clique number and a natural…

The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and…

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all…

We consider a problem of approximating the size of the largest clique in a graph, with a monotone circuit. Concretely, we focus on distinguishing a random Erd\H{o}s-Renyi graph $\mathcal{G}_{n,p}$, with $p=n^{-\frac{2}{\alpha-1}}$ chosen…

We prove that every $k$-list-critical graph ($k \ge 7$) on $n \ge k+2$ vertices has at least $\frac12 \left(k-1 + \frac{k-3}{(k-c)(k-1) + k-3}\right)n$ edges where $c = (k-3)\left(\frac12 - \frac{1}{(k-1)(k-2)}\right)$. This improves the…

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of…

We provide a finite forbidden induced subgraph characterization for the graph class $\varUpsilon_k$, for all $k \in \mathbb{N}_0$, which is defined as follows. A graph is in $\varUpsilon_k$ if for any induced subgraph, $\Delta \leq \chi -1…

We introduce the $k$-variable-occurrence fragment, which is the set of terms having at most $k$ occurrences of variables. We give a sufficient condition for the decidability of the equational theory of the $k$-variable-occurrence fragment…

In this paper we consider the fundamental problem of finding subgraphs in highly dynamic distributed networks - networks which allow an arbitrary number of links to be inserted / deleted per round. We show that the problems of $k$-clique…

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind…

This paper shows how to harness existing theorem provers for first-order logic to automatically verify safety properties of imperative programs that perform dynamic storage allocation and destructive updating of pointer-valued structure…

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can…

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with modular predicates. Our approach aims toward the most generic statements that we could achieve, which…

Given a graph $G$ and a parameter $k$, the $k$-biclique problem asks whether $G$ contains a complete bipartite subgraph $K_{k,k}$. This is the most easily stated problem on graphs whose parameterized complexity is still unknown. We provide…

Siran constructed infinite families of k-crossing-critical graphs for every k=>3 and Kochol constructed such families of simple graphs for every k=>2. Richter and Thomassen argued that, for any given k>=1 and r>=6, there are only finitely…

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold $\lambda_c$ for the emergence of a non-trivial k-core in the random graph $G(n,\lambda/n)$, and…

We study logical limit laws for preferential attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $1$, we start with vertices $0,1$ and $m$ edges between them. At step $n+1$ the vertex…

Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…