Related papers: Large monochromatic components in hypergraphs with…

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex…

Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $\chi(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of…

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex…

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…

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 say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number…

Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$, the ($d,kn$)-differential coloring problem asks for a coloring of the vertices of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as…

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 $k$…

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a…

We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\Omega(\sqrt{n})$. This improves on the…

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive…

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such…

An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of…

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 $k$-partite $k$-graph with $n$ vertices in each partition class, and let $\delta_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize those $H$ with $\delta_{k-1}(H) \geq n/2$ and with no perfect matching. As a…

In 1973 P. Erd\H{o}s and L. Lov\'asz noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erd\H{o}s and Lov\'asz proved that…

Let $\mathcal{H}$ be a hypergraph of maximal vertex degree $\Delta$, such that each its hyperedge contains at least $\delta$ vertices. Let $k=\lceil\frac{2\Delta}{\delta}\rceil$. We prove that (i) The hypergraph $\mathcal{H}$ admits proper…

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every…