Related papers: Maximal Independent Sets in Planar Triangulations

For a planar point-set $P$, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in $P$ lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and…

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials,…

We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…

Given a finite point set $X$ in the plane, the degree of a pair $\{x,y\} \subset X$ is the number of empty triangles $t=conv\{x,y,z\}$, where empty means $t\cap X=\{x,y,z\}$. Define $deg X$ as the maximal degree of a pair in $X$. Our main…

Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set…

By Brook's Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs…

In this paper, connections between independent sets and the variety of mutual-visibility sets are studied. It is proved that every outer mutual-visibility set of a graph is independent if and only if the graph is distance edge-critical.…

Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the…

An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an…

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of…

We introduce a new framework for reconfiguration problems, and apply it to independent sets as the first example. Suppose that we are given an independent set $I_0$ of a graph $G$, and an integer $l \ge 0$ which represents a lower bound on…

Given a finite, simple graph $G$, the independent bondage number of $G$ is the minimum size of an edge set such that its deletion results in a graph with strictly larger independent domination number than that of $G$. While the bondage…

We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the…

In this work we study the fencing problem consisting of finnding a trisection of a 3-rotationally symmetric planar convex body which minimizes the maximum relative diameter. We prove that an optimal solution is given by the so-called…

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…

Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $R^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\geq \Omega(n\sqrt{\log n})$, which answers a question…

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n =…

Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct…