Related papers: A note on equitable colorings of forests
This paper derives a unifying theorem establishing consistency results for a broad class of tree-based algorithms. It improves current results in two aspects. First of all, it can be applied to algorithms that vary from traditional Random…
We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles,…
The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the…
In this note, we show various versions of holomorphic Morse inequalities tensoring with a coherent sheaf.
We study various types of consistency of honest decision trees and random forests in the regression setting. In contrast to related literature, our proofs are elementary and follow the classical arguments used for smoothing methods. Under…
In this note, we prove the regularity of eta forms by the Clifford asymptotics. Then we generalize this result to the equivariant case.
We prove uniform consistency of Random Survival Forests (RSF), a newly introduced forest ensemble learner for analysis of right-censored survival data. Consistency is proven under general splitting rules, bootstrapping, and random selection…
A result about spanning forests for graphs yields a short proof of Krebes's theorem concerning embedded tangles in links.
This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same $U$-polynomial (or, equivalently, the same chromatic symmetric function). We consider the $U_k$-polynomial, which is a…
Consider the following process on a simple graph without isolated vertices: Order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a…
For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We…
The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…
A graph in which all minimal zero forcing sets are in fact minimum size is called ``well-forced." This paper characterizes well-forced trees and presents an algorithm for determining which trees are well-forced. Additionally, we…
For a graph G = (V,E) where each vertex is coloured by one of k colours, consider a subset C of V such that for each vertex v in V\C, its set of nearest neighbours in C contains at least one vertex of the same colour as v. Such a C is…
We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and…
An instance of colorful k-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius \r{ho} such that there exist…
The purpose of this note is to give a self contained description of Walls finiteness obstruction.
This note is the follow up to a paper by M. Waldschmidt.
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic threshold of a graph $G$, denoted by $\chi_=^*(G)$, is the minimum $k$ such that $G$ is equitably…
In this short paper we show that the inequality of arithmetic and geometric means is reduced to another interesting inequality, and a proof is provided.