Related papers: $k$-colored kernels in semicomplete multipartite d…

We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed…

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can…

For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a rainbow…

Let $k$ be an integer with $k\geq 2$. A $k$-king in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$ and a non-king is a vertex which is not a 3-king. A subset $K$ is $k$-independent if…

Let $D$ be a digraph. We call a subset $N$ of $V(D)$ $k$-independent if for every pair of vertices $u,v \in N$, $d(u,v) \geq k$; and we call it $\ell$-absorbent if for every vertex $u \in V(D) \setminus N$, there exists $v \in N$ such that…

Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if…

In this paper, we introduce the concept of bikernel by monochromatic paths of a bicolored digraph. This concept is strongly motivated by the existing notions of kernels, kernels by monochromatic paths, and double stable augmented…

Let $D = (V(D), A(D))$ be a digraph. A subset $S \subseteq V(D)$ is $k$-independent if the distance between every pair of vertices of $S$ is at least $k$, and it is $\ell$-absorbent if for every vertex $u$ in $V(D) \setminus S$ there exists…

An edge-coloring of a hypergraph is {\em spanning} if every vertex sees every color used in the coloring. In this paper, we prove that for $k \geq 2r \geq 6$, in any spanning $k$-coloring of the edges of a complete $r$-partite $r$-uniform…

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{m,n}$ there is a monochromatic connected component with at least ${m+n\over r}$ vertices. In this paper we study an extension of this problem by…

A directed graph $D=(V(D),A(D))$ has a kernel if there exists an independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)-K$ has an ingoing arc $u\mathbin{\longrightarrow}v$ for some $u\in K$. There are directed graphs that do…

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that…

Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for…

A {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same…

In 2018, Bai, Fujita and Zhang (\emph{Discrete Math.} 2018, 341(6): 1523-1533) introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph $D$, which is a subset $S$ of vertices of $D$ such that…

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a…

Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$.…

Let $n, k, m$ be positive integers with $n\gg m\gg k$, and let $\mathcal{A}$ be the set of graphs $G$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

In this paper, we introduce the concept of up-color kernel, which is a generalization of a kernel for vertex-colored digraphs. We give sufficient and necessary conditions for several families of digraphs to have an up-color kernel, as well…