Related papers: Extremal results for graphs avoiding a rainbow sub…

There has been extensive studies on the following question: given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a `colorful' copy of $H$, i.e., a copy of $H$ containing at most one edge…

Let $\mathcal{G}_n^k=\{G_1,G_2,\ldots,G_k\}$ be a multiset of graphs on vertex set $[n]$ and let $F$ be a fixed graph with edge set $F=\{e_1, e_2,\ldots, e_m\}$ and $k\ge m$. We say ${\mathcal{G}_n^k}$ is rainbow $F$-free if there is no…

Let F,G,H be three graphs on the same n vertices. We consider the maximum of the sum and product of the number of their edges subject to the condition in the title.

In this paper we show that for $r\geq 12$ and any sufficiently large $n$-vertex graph $G$ the number of $r$-edge-colorings of $G$ with no rainbow $K_4$ is at most $r^{ex(n,K_4)}$, where $ex(n,K_4)$ denotes the Tur\'{a}n number of $K_4$.…

In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$…

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…

Mantel's Theorem asserts that a simple $n$ vertex graph with more than $\frac{1}{4}n^2$ edges has a triangle (three mutually adjacent vertices). Here we consider a rainbow variant of this problem. We prove that whenever $G_1, G_2, G_3$ are…

Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow…

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for…

In an edge-colored graph $G$, a rainbow clique $K_k$ is a $k$-complete subgraph in which all the edges have distinct colors. Let $e(G)$ and $c(G)$ be the number of edges and colors in $G$, respectively. In this paper, we show that for any…

Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the…

Let $G$ be an edge-colored graph on $n$ vertices. For a vertex $v$, the \emph{color degree} of $v$ in $G$, denoted by $d^c(v)$, is the number of colors appearing on the edges incident with $v$. Denote by $\delta^c(G)=\min\{d^c(v):v\in…

A famous conjecture of Caccetta and H\"aggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ there is a directed cycle of length $r$ or less. We consider the following generalization: in an undirected…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Tur\'an number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with…

Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on the same vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a…

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…

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…

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$…

For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors.…