Related papers: Rainbow independent sets in certain classes of gra…

For a given class ${\cal C}$ of graphs and given integers $m \le n$, let $f_{\cal C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to ${\cal C}$ have a (possibly partial) rainbow independent…

Given a family $\mathcal{I}$ of independent sets in a graph, a rainbow independent set is an independent set $I$ such that there is an injection $\phi\colon I\to \mathcal{I}$ where for each $v\in I$, $v$ is contained in $\phi(v)$. Aharoni,…

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $\mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019)…

For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a…

We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of…

For a family of graphs $\cF$, a graph is called $\cF$-free if it does not contain any member of $\cF$ as a subgraph. Given a collection of graphs $(G_1,\ldots,G_t)$ on the same vertex set $V$ of size $n$, a rainbow graph on $V$ is obtained…

A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by…

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper, generalizes these problems to edge colored graphs with given total color degree. In…

In this paper, we present results for the rainbow neighbourhood numbers of set-graphs. It is also shown that set-graphs are perfect graphs. The intuitive colouring dilemma in respect of the rainbow neighbourhood convention is clarified as…

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of…

Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such…

Let $X_1,X_2,\ldots,X_n$ be chosen independently and uniformly at random from the unit $d$-dimensional cube $[0,1]^d$. Let $r$ be given and let $\cal X=\{X_1,X_2,\ldots,X_n\}$. The random geometric graph $G=G_{\cal X,r}$ has vertex set…

Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…

For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in…

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of…

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…

Let $S$ be a set of $n$ points in general position in the plane. Suppose that each point of $S$ has been assigned one of $k \ge 3$ possible colors and that there is the same number, $m$, of points of each color class. A polygon with…

Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a…

An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $\alpha$.…