Related papers: A probabilistic Hadwiger-Nelson problem
If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…
The number of proper $q$-colorings of a graph $G$, denoted by $P_G(q)$, is an important graph parameter that plays fundamental role in graph theory, computational complexity theory and other related fields. We study an old problem of Linial…
We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollob\'as, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$…
We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same…
Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…
Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the…
Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$, the ($d,kn$)-differential coloring problem asks for a coloring of the vertices of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as…
For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $\chi'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges…
We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with…
Let $G$ be a graph on $n$ vertices and let $k$ be a fixed positive integer. We denote by $\mathcal G_{\text{$k$-out}}(G)$ the probability space consisting of subgraphs of $G$ where each vertex $v\in V(G)$ randomly picks $k$ neighbors from…
We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom,…
The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring,…
Dense subgraph discovery methods are routinely used in a variety of applications including the identification of a team of skilled individuals for collaboration from a social network. However, when the network's node set is associated with…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform…
An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…
We study the max k-cut game on an undirected and unweighted graph in order to find out whether an optimal solution is also a strong equilibrium. While we do fail to show that, by proving an alternate formula for computing the cut value…
K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite…