Related papers: Blow-up trick in Combinatorics

An overlap representation is an assignment of sets to the vertices of a graph in such a way that two vertices are adjacent if and only if the sets assigned to them overlap. The overlap number of a graph is the minimum number of elements…

This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the…

We introduce a graph decomposition which exists for all simple, connected graphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that each vertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is `balanced':…

Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on…

Graphs are a natural representation for systems based on relations between connected entities. Combinatorial optimization problems, which arise when considering an objective function related to a process of interest on discrete structures,…

The generalized Oberwolfach problem asks for a decomposition of a graph $G$ into specified 2-regular spanning subgraphs $F_1,\ldots, F_k$, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are…

Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of…

The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized…

Graph burning models the spread of information or contagion in a graph. At each time step, two events occur: neighbours of already burned vertices become burned, and a new vertex is chosen to be burned. The big conjecture is known as the…

Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition.…

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…

The $k$-cap (or $k$-winners-take-all) process on a graph works as follows: in each iteration, exactly $k$ vertices of the graph are in the cap (i.e., winners); the next round winners are the vertices that have the highest total degree to…

The paper deals with blow--up for the solutions of wave equation with nonlinear source and nonlinear boudary damping terms, posed in a bounded and regular domain. The initial data are posed in the energy space. The aim of the paper is to…

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…

Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of…

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The…

In this talk we discuss sector decomposition. This is a method to disentangle overlapping singularities through a sequence of blow-ups. We report on an open-source implementation of this algorithm to compute numerically the Laurent…

Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of…

Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.