Related papers: Maximum-Width Rainbow-Bisecting Empty Annulus
We consider the problem of computing the largest-area bichromatic separating box among a set of $n$ red points and a set of $m$ blue points in three dimensions. Currently, the best-known algorithm to solve this problem takes $O(m^2 (m +…
The input to the no-rainbow hypergraph coloring problem is a hypergraph $H$ where every hyperedge has $r$ nodes. The question is whether there exists an $r$-coloring of the nodes of $H$ such that all $r$ colors are used and there is no…
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…
Given a set of points $P$ and axis-aligned rectangles $\mathcal{R}$ in the plane, a point $p \in P$ is called \emph{exposed} if it lies outside all rectangles in $\mathcal{R}$. In the \emph{max-exposure problem}, given an integer parameter…
A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then.…
Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform…
We study a new variant of colored orthogonal range searching problem: given a query rectangle $Q$ all colors $c$, such that at least a fraction $\tau$ of all points in $Q$ are of color $c$, must be reported. We describe several data…
In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with…
Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the…
Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to…
Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors,…
For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different…
Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a…
We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the…
The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It was recently proved that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the…
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which…
Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number…
An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph $H$, the rainbow Tur\'an number $\mathrm{ex}^{\ast}(n,H)$ is defined as the maximum number of edges in a properly edge-colored graph on…
In this paper we analyze the length-spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this…