Related papers: A different short proof of Brooks' theorem

In this expository note, we discuss an early partial coloring result of B. Kashin [C. R. Acad. Bulgare Sci., 1985]. Although this result only implies Spencer's six standard deviations [Trans. Amer. Math. Soc., 1985] up to a $\log\log n$…

Let $G$ be a simple undirected graph on $n$ vertices with maximum degree~$\Delta$. Brooks' Theorem states that $G$ has a $\Delta$-colouring unless~$G$ is a complete graph, or a cycle with an odd number of vertices. To recolour $G$ is to…

We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out…

In this note, we will give a short proof of an identity for cubic partitions.

The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…

We study \L o\'s's theorem in a choiceless context. We introduce some variants of \L o\'s's theorem. These variants seem weaker than \L o\'s's theorem, but we prove that these are equivalent to \L o\'s's theorem.

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound $\chi(G) \leq (1+o(1))\Delta(G)/\ln \Delta(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy…

We consider the problem of online graph colouring. Whenever a node is requested, a colour must be assigned to the node, and this colour must be different from the colours of any of its neighbours. According to the greedy algorithm the node…

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous…

We show that the Kneser graph of triangulations of a convex $n$-gon has chromatic number $n-2$.

We give a pictorial proof that transparently illustrates why four colours suffce to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show,…

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then…

Most of the assertions in the theory of well ordered sets are quite simple. However, one of its central statements, Zermelo's theorem, stands out of this rule, for its well-known proofs are rather complicated. The aim of the current paper…

In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques from model theory to prove the function field version of the conjecture. In doing so he was able to answer a related question of Voloch, which we refer to…

In this note we provide a quick proof of the Sklar's Theorem on the existence of copulas by using the generalized inverse functions as in the one dimensional case, but a little more sophisticated.

We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory…

In this paper, we introduce the notion of "$Lucas-Coloring$" associated with a planar graph $g$. When $g$ is a $4$-regular, the enumeration of $Lucas-Coloring$ has an interesting interpretation. Specifically, it yields a numerical invariant…

We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…

A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which…