Related papers: Conflict-Free Coloring Made Stronger

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any…

Let $2\le k\in\mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an {\it efficient total coloring} if each color yields an efficient dominating set, where the efficient domination condition applies to the…

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the…

A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $\Delta$, sampling $O(\log{n})$ colors per each vertex independently from $\Delta+1$ colors almost…

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…

We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored…

There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, in the \emph{entire coloring} of a plane graph we are to color these three sets so that any pair…

We consider the single-conflict coloring problem, a graph coloring problem in which each edge of a graph receives a forbidden ordered color pair. The task is to find a vertex coloring such that no two adjacent vertices receive a pair of…

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show…

A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a…

We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where…

In the pliable index coding (PICOD) problem, a server is to serve multiple clients, each of which possesses a unique subset of the complete message set as side information and requests a new message which it does not have. The goal of the…

A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL…

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to…

For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a…

We prove that any finite set of half-planes can be colored by two colors so that every point of the plane, which belongs to at least three half-planes in the set, is covered by half-planes of both colors. This settles a problem of Keszegh.

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to…

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we…

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A…