Related papers: Vertex Colorings without Rainbow or Monochromatic …
In this paper we study a problem of vertex two-coloring of undirected graph such that there is no monochromatic cycle of given length. We show that this problem is hard to solve. We give a proof by presenting a reduction from variation of…
For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous…
The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Gr\"unbaum (1973) together with the acyclic coloring of…
Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure.…
We popularize the question whether, for $m$ large enough, all $m$-uniform shift-chain hypergraphs are properly $2$-colorable. On the other hand, we show that for every $m$ some $m$-uniform shift-chains are not polychromatic $3$-colorable.
For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short),…
Let $G$ be a nontrivial connected and vertex-colored graph. A subset $X$ of the vertex set of $G$ is called rainbow if any two vertices in $X$ have distinct colors. The graph $G$ is called \emph{rainbow vertex-disconnected} if for any two…
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…
A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable $k$-uniform hypergraph, it is…
For a graph with colored vertices, a rainbow subgraph is one where all vertices have different colors. For graph $G$, let $c_k(G)$ denote the maximum number of different colors in a coloring without a rainbow path on $k$ vertices, and…
The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine…
For a given $\delta \in (0,1)$, the randomly perturbed graph model is defined as the union of any $n$-vertex graph $G_0$ with minimum degree $\delta n$ and the binomial random graph $\mathbf{G}(n,p)$ on the same vertex set. Moreover, we say…
We say a graph $H$ is $r$-rainbow-uncommon if the maximum number of rainbow copies of $H$ under an $r$-coloring of $E(K_n)$ is asymptotically (as $n \to \infty$) greater than what is expected from uniformly random $r$-colorings. Via…
The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper, generalizes these problems to edge colored graphs with given total color degree. In…
Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to…
A well known problem from an excellent book of Lov\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh…
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some…
Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors,…
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph…