Related papers: Colored ray configurations
We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its…
We study algorithmic matroid intersection coloring. Given $k$ matroids on a common ground set $U$ of $n$ elements, the goal is to partition $U$ into the fewest number of color classes, where each color class is independent in all matroids.…
We prove a formula for the asymptotic number of edge-colored regular graphs with a prescribed set of allowed vertex-incidence structures. The formula depends on specific critical points of a polynomial encoding the vertex-incidences. As an…
We consider the enumeration of plane trees (rooted ordered trees) whose vertices are colored according to a specific coloring rule that prescribes which possible pairs of colors can occur as the colors of a parent vertex and its child. This…
A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted…
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced…
Given a geometric hypergraph (or a range-space) $H=(V,\cal E)$, a coloring of its vertices is said to be conflict-free if for every hyperedge $S \in \cal E$ there is at least one vertex in $S$ whose color is distinct from the colors of all…
Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in…
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the…
Let $P$ be a finite set of points in general position in the plane. The disjointness graph of segments $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in…
Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of $3$-dimensional parallelepiped arrangements. In the case each parallelepiped is within one floor, a direct application of the…
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number…
We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…
Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there…
We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in…
In this paper, we consider the problem of identifying patterns of interest in colored strings. A colored string is a string where each position is assigned one of a finite set of colors. Our task is to find substrings of the colored string…
The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…
Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite…
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will…
An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a…