相关论文: The power of many colours

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},…

A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a…

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does…

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority…

Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $\Omega\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to…

We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of…

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices.…

A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This paper studies clustered colorings of graphs,…

The 2-colorable perfect matching problem asks whether a graph can be colored with two colors so that each node has exactly one neighbor with the same color as itself. We prove that this problem is NP-complete, even when restricted to…

An $r$-uniform hypergraph $H = (V, E)$ is $r$-partite if there exists a partition of the vertex set into $r$ parts such that each edge contains exactly one vertex from each part. We say an independent set in such a hypergraph is balanced if…

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of $b$-simple hypergraphs, in which any two edges do not share more than $b$ common vertices. We prove that for…

There are two possible definitions of the "s-disjoint r-uniform Kneser hypergraph'' of a set system T: The hyperedges are either r-sets or r-multisets. We point out that Ziegler's (combinatorial) lower bound on the chromatic number of an…

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors $\chi$. Each cell $s$ in the grid is assigned a subset of colors $\chi_s \subseteq \chi$ and should be…

For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to $\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For…

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…

The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of…

Extremal problems concerning the number of complete subgraphs have a long story in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in a graph $G$ and $m={{r_m}\choose s}+t_m$, where $0\le t_m\leq r_m$. Edr\H{o}s showed that…

A balanced colouring of a graph is one in which every colour appears the same number of times. Given a fixed graph $H$ on $r$ vertices and a balanced $k$-colouring of the complete graph $K_{nrk}$, Hollom (2025) asked the following question:…