Related papers: Rainbow saturation of graphs
An edge-coloured path is rainbow if all of its edges have distinct colours. Let $G$ be a connected graph. The rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any…
An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the…
Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous…
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…
Let $f(n,r,k)$ be the minimal number such that every hypergraph larger than $f(n,r,k)$ contained in $\binom{[n]}{r}$ contains a matching of size $k$, and let $g(n,r,k)$ be the minimal number such that every hypergraph larger than $g(n,r,k)$…
In 2017, Ron Aharoni made the following conjecture about rainbow cycles in edge-coloured graphs: If $G$ is an $n$-vertex graph whose edges are coloured with $n$ colours and each colour class has size at least $r$, then $G$ contains a…
The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich…
A famous conjecture of Caccetta and H\"{a}ggkvist (CHC) states that a directed graph $D$ with $n$ vertices and minimum outdegree at least $r$ has a directed cycle of length at most $\lceil \frac{n}{r}\rceil$. In 2017, Aharoni proposed the…
A {\it rainbow matching} in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(\delta) such that a properly edge-colored graph G with minimum degree \delta and order at least…
An edge-coloured path is rainbow if its edges have distinct colours. For a connected graph $G$, the rainbow connection number (resp. strong rainbow connection number) of $G$ is the minimum number of colours required to colour the edges of…
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 $rc(G)$, is the smallest number of colors that are…
We study the following problem. How many distinct copies of $H$ can an $n$-vertex graph $G$ have, if $G$ does not contain a rainbow $F$, that is, a copy of $F$ where each edge is contained in a different copy of $H$? The case $H=K_r$ is…
The $k$-rainbow independent domination number of a graph $G$, denoted $\gamma_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus…
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…
Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in…
The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a…
Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein…
A natural question, inspired by the famous Ryser-Brualdi-Stein Conjecture, is to determine the largest positive integer $g(r,n)$ such that every collection of $n$ matchings, each of size $n$, in an $r$-partite $r$-uniform hypergraph…
The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…
The rainbow Tur\'an number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$,…