Related papers: Rainbow matchings and algebras of sets
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic…
Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two…
For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$…
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…
A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most…
For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy…
A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as…
A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets…
The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is…
An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to…
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…
DeVos et al conjectured that if $M$ is a simple, regular matroid and $c$ is a colouring of the elements of $M$ with $r(M)+1$ colours, where each colour class has at least two elements, then $M$ contains a rainbow circuit of size at most…
We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\textbf{G}=\{G_1, G_2,\ldots,…
For an $r$-graph $F$ and integers $n,t$ satisfying $t \le n/v(F)$, let $\mathrm{ar}(n,tF)$ denote the minimum integer $N$ such that every edge-coloring of $K_{n}^{r}$ using $N$ colors contains a rainbow copy of $tF$, where $tF$ is the…
A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$,…
Given an edge-colored complete graph $K_n$ on $n$ vertices, a perfect (respectively, near-perfect) matching $M$ in $K_n$ with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we…
Let $\mathcal F_1,\ldots, \mathcal F_s\subset [n]^k$ be a collection of $s$ families. In this paper, we address the following question: for which sequences $f_1,\ldots, f_s$ the conditions $|\ff_i|>f_i$ imply that the families contain a…
A path in an edge-colored graph is called a rainbow path if every two distinct edges of the path have different colors. A graph whose every pair of vertices are linked by a rainbow path is called a rainbow-connected graph. The rainbow…
A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$…
An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that…