Related papers: Kernels by properly colored paths in arc-colored d…

Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of…

We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph…

Let $D$ be a digraph. A stable set $S$ of $D$ and a path partition $\mathcal{P}$ of $D$ are orthogonal if every path $P \in \mathcal{P}$ contains exactly one vertex of $S$. In 1982, Berge defined the class of $\alpha$-diperfect digraphs. A…

In a digraph, a kernel is a subset of vertices that is both independent and absorbing. Kernels have important applications in combinatorics and outside. Kernels do not always exist and finding sufficient conditions ensuring their existence…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

An edge-coloured cycle is rainbow if the edges have distinct colours. Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required…

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been…

A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper…

A directed graph $D$ is singly connected if for every ordered pair of vertices $(s,t)$, there is at most one path from $s$ to $t$ in $D$. Graph orientation problems ask, given an undirected graph $G$, to find an orientation of the edges…

Given an undirected graph $G=(V,E)$, vertices $s,t\in V$, and an integer $k$, Tracking Shortest Paths requires deciding whether there exists a set of $k$ vertices $T\subseteq V$ such that for any two distinct shortest paths between $s$ and…

A \emph{geometric graph} is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \ge 2$, there exists a constat $c>0$…

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…

We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $\omega(H)$ many colors where $\omega(H)$ is the clique number of $H$. A graph is perfectly colorable…

With the recent rise in the amount of structured data available, there has been considerable interest in methods for machine learning with graphs. Many of these approaches have been kernel methods, which focus on measuring the similarity…

Graph kernels are usually defined in terms of simpler kernels over local substructures of the original graphs. Different kernels consider different types of substructures. However, in some cases they have similar predictive performances,…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

Let $\vec{K}_{\mathbb{N}}$ 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 $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has…

Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring…

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called…

Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S \in E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. The discrete interval…