Related papers: A short proof of the Four Colour Theorem

In this paper, we introduce the notion of F-manifold color algebras and study their properties which extend some results for F-manifold algebras.

Van der Waerden's (VDW) colouring theorem in combinatoric number theory [1] has scope for physical applications.The solution of the two colour case has enabled the construction of an explicit mapping of an infinite, one dimensional…

In a recent paper, Dousse introduced a refinement of Siladi\'c's theorem on partitions, where parts occur in two primary and three secondary colors. Her proof used the method of weighted words and $q$-difference equations. The purpose of…

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on…

The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…

For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$. The block-count…

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof…

The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the…

We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list…

Let $Q_d$ be the $d$-dimensional hypercube and $N=2^d$. We prove that the number of (proper) 4-colorings of $Q_d$ is asymptotically \[6e2^N,\] as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information…

We give a short proof of Ahlfors' theorem on covering surfaces.

In the paper we prove, in particular, that for any measurable coloring of the euclidian plane into two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields…

This paper discusses reformulations of the problem of coloring plane maps with four colors. The context is the edge-coloring with three colors of cubic graphs such that three distinct colors occur at each vertex. We include discussion of…

We provide a human-verifiable proof that, in a certain sense, the chromatic number of the plane is exactly 7.

We sketch several proofs of F\'ary--Milnor theorem.

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the…

In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions.…

This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Gr\"unbaum colorings, their existence and usage. In particular, one of the most…

We give a simpler proof of Seymour's Theorem on edge-coloring series-parallel multigraphs and derive a linear-time algorithm to check whether a given series-parallel multigraph can be colored with a given number of colors.

We define "paradoxical colouring rule", show its relation to measure theoretic paradoxes, and demonstrate that proper vertex colouring can be a paradoxical colouring rule.