Related papers: A manual proof of the Four-colour Theorem

Proving for triangulations an extended version of the 4-colour theorem by induction, we manage to exclude the case which led to the failure of Kempe's attempted proof. The new idea is to claim the existence of a "nice" 4-colouring, in which…

We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT).…

In RSST, they "replace the mammoth hand-checking of unavoidability that A&H required, by another mammoth hand-checkable proof " (page 18). Here, the proof of unavoidability is accomplished in a lengthy structured hand-checkable proof whose…

The four-color conjecture has puzzled mathematicians for over 170 years and has yet to be proven by purely mathematical methods. This series of articles provides a purely mathematical proof of the four-color conjecture, consisting of two…

We provide a simple proof of Kamp's theorem.

We present in this work a new and simple proof of the false centre theorem.

The approach is through a singularity analysis of generating functions for 3- and 4-connected triangulations, asymptotic analysis, properties of the ${{}_3F_2}$ hypergeometric series, and Tutte's enumerative work on planar maps and…

The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such…

We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points unit distance apart which are identically colored.

We prove a better coloring theorem for aleph_4 and even aleph_3. This has a general topology consequence.

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

In this survey essay, I explore the application of the discharging method in graph theory, including the selection of charging rules and discharging rules, and the general characteristics of the discharging method. As examples, I will prove…

I present a simple, elementary proof of Morley's theorem, highlighting the naturalness of this theorem.

Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non trivial model soluble without computer. Four colors are necessary and sufficient.

An approach of using RGB-tilings for proving the Four Color Theorem discussed in three previous work is expanded in this paper. A novel methodology and revisions for the methodology in the three aforementioned papers are discussed, and a…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…

The main result of this paper is a "colored Tverberg theorem for rainbow-unavoidable complexes". This theorem may be considered as a merging of two theorems: "Tverberg theorem for collectively unavoidable complexes" and "balanced colored…

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs…

In this note, we present a simple directed graph proof of Sharkovsky's theorem.

We present a proof of the Moon in a puddle theorem, and use its key lemma to prove a generalization of the four-vertex theorem.