Related papers: New results relating independence and matchings

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double…

We take a different look at the problem of testing the independence of two metric-space-valued random variables using the distance correlation. Instead of testing if the distance correlation vanishes exactly, we are interested in the…

Let $N\geq 3$ be a composite, odd, and square-free integer and let $\Gamma$ be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular curve of genus $g_{\Gamma}$ associated to $\Gamma$. In this article, we study the…

Charge independence and symmetry are approximate symmetries of nature. The observations of the small symmetry breaking effects and the consequences of those effects are reviewed. The effects of the mass difference between up and down quarks…

We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be…

Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix of a digraph $G$ is defined as $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$$ where $A(G)$ is the adjacency matrix of…

A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a…

A matching $M$ in a graph $G$ is an \emph{acyclic matching} if the subgraph of $G$ induced by the endpoints of the edges of $M$ is a forest. Given a graph $G$ and a positive integer $\ell$, Acyclic Matching asks whether $G$ has an acyclic…

For $\mu, \kappa$ infinite, say $\mathcal{A}\subseteq [\kappa]^\kappa$ is a $(\mu,\kappa)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<\mu}$ then…

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$.…

For a set of dependent random variables, without stationary or the strong mixing assumptions, we derive the asymptotic independence between their sums and maxima. Then we apply this result to high-dimensional testing problems, where we…

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $\alpha(G)$. More precisely, we prove that every digraph $G$ has…

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge…

We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $\Delta$ on $n$ vertices, for all $k \le (1-\delta)m^*(G)$, where $\delta>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an…

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by…

The power graph $P(G)$ of a finite group $G$ is the undirected simple graph with vertex set $G$, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are…

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from…

Let $G$ be a graph on $n$ vertices of maximum degree $\Delta$. We show that, for any $\delta > 0$, the down-up walk on independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$ mixes in time $O_{\Delta,\delta}(k\log{n})$, thereby…