Related papers: Monochromatic components with many edges
It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can…
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete…
A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and…
Given a $k$-colouring of the edges of the complete graph $K_n$, are there $k-1$ monochromatic components that cover its vertices? This important special case of the well-known Lov\'asz-Ryser conjecture is still open. In this paper we…
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…
We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint…
We prove that every graph with circumference at most $k$ is $O(\log k)$-colourable such that every monochromatic component has size at most $O(k)$. The $O(\log k)$ bound on the number of colours is best possible, even in the setting of…
We consider the following question of Bollobas: given an r-colouring of the edges of the complete graph on n vertices, how large a k-connected subgraph can we find using only one colour? We solve this problem asymptotically when r-1 is a…
We study two variations of the Gyarfas--Lehel conjecture on the minimum number of monochromatic components needed to cover an edge-coloured complete bipartite graph. Specifically, we show the following. - For p>> (\log n/n)^{1/2},…
An edge-coloring of the complete graph $K_n$ we call $F$-caring if it leaves no $F$-subgraph of $K_n$ monochromatic and at the same time every subset of $|V(F)|$ vertices contains in it at least one completely multicolored version of $F$.…
Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$ let $f(H,c)$ be the number of components in the subhypergraph induced by the color class with the least number of components. Let $f_k(H)$ be the maximum possible…
An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$…
Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that…
An edge-coloring of a connected graph $G$ is called a {\it monochromatic connection coloring} (MC-coloring, for short), introduced by Caro and Yuster, if there is a monochromatic path joining any two vertices of the graph $G$. Let $mc(G)$…
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and…
We study a generalization of a famous result of Goodman and establish that asymptotically at least a $1/256$ fraction of all triangles needs to be monochromatic in any four-coloring of the edges of a complete graph. We also show that any…
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…
The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a…
For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides $n$ and an affine plane of order $r$ exists, we construct an $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ such that the largest monochromatic…
We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erd\H{o}s. We also prove a corresponding stability result, showing that…