Related papers: On the $d$-independence number in 1-planar graphs
The interval number of a graph $G$ is the minimum $k$ such that one can assign to each vertex of $G$ a union of $k$ intervals on the real line, such that $G$ is the intersection graph of these sets, i.e., two vertices are adjacent in $G$ if…
An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…
A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…
A theorem of Shearer states that every $n$-vertex triangle-free graph of maximum degree $d \geq 2$ contains an independent set of size at least $(d\log d - d + 1)/(d - 1)^2 \cdot n$. Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di…
The maximum number of edges in a graph with matching number m and maximum degree d has been determined in [1] and [2], where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is…
The domatic number of a graph $G$ is the maximum number of pairwise disjoint dominating sets of $G$. We are interested in the LP-relaxation of this parameter, which is called the fractional domatic number of $G$. We study its extremal value…
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the…
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph…
An independent set $I$ in a graph $G$ is maximal if $I$ is not properly contained in any other independent set of $G$. The study of maximal independent sets (MIS's) in various graphs is well-established, often focusing upon enumeration of…
We describe an infinite family of graphs $G_n$, where $G_n$ has $n$ vertices, independence number at least $n/4$, and no set of less than $\sqrt{n}/2$ vertices intersects all its maximum independent sets. This is motivated by a question of…
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We bound the maximum modulus, $\mbox{maxmod}(n)$, of an independence root over…
Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, and mu(G) is the size of a maximum matching. The number…
The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…
Let $G$ be a simple graph with vertex set $V\left( G\right) $. A set $S\subseteq V\left( G\right) $ is independent if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$. The…
A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$.…
Let $G=(V,E)$ be a finite simple graph. In this paper, we study the degree of the $h$-polynomial of the edge ideal of $G$ in relation to the independence number of $G$. Our approach is based on the value of the independence polynomial of…
Consider the problem of determining the maximal induced subgraph in a random $d$-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large $d$, that any such…
Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The…
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…
In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph…