Related papers: Monochromatic Edges in Complete Multipartite Hyper…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

A road interchange where $n$ roads meet and in which the drivers are not allowed to change lanes can be modelled as an embedding of a 2-coloured (hence bipartite) multigraph $G$ with equal-sized colour classes into an orientable surface…

Let the diameter cover number, $D^t_r(G)$, denote the least integer $d$ such that under any $r$-coloring of the edges of the graph $G$, there exists a collection of $t$ monochromatic subgraphs of diameter at most $d$ such that every vertex…

The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform…

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a…

We show that if a multigraph $G$ with maximum edge-multiplicity of at most $\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices, then it…

We show that for every $\eta>0$, there exists $n_0$ such that for every even $n$, $n\ge n_0$, and every graph $G$ with $(2+\eta)n$ vertices and minimum degree at least $(7/4+4\eta)n$, each colouring of the edges of $G$ with three colours…

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order…

A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph…

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…

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength}…

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit.…

In this paper we study the minimal size of edges in hypergraph families that guarantees the existence of a polychromatic coloring, that is, a $k$-coloring of a vertex set such that every hyperedge contains a vertex of all $k$ color classes.…

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the…

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$…

We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one…

There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to…

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…

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…

For a given hypergraph $H = (V,E)$ consider the sum $q(H)$ of $2^{-|e|}$ over $e \in E$. Consider the class of hypergraphs with the smallest edge of size $n$ and without a 2-colouring without monochromatic edges. Let $q(n)$ be the smallest…