Related papers: A pictorial proof of the Four Colour Theorem

A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete…

The four-color theorem states that no more than four colors are required to color all nodes in planar graphs such that no two adjacent nodes are of the same color. The theorem was first propounded by Francis Guthrie in 1852. Since then,…

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to…

For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to…

In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be…

A formal proof has not been found for the four color theorem since 1852 when Francis Guthrie first conjectured the four color theorem. Why? A bad idea, we think, directed people to a rough road. Using a similar method to that for the formal…

The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976…

This paper presents a path to proving the Four-Color Theorem that differs from the traditional "reducible configuration" method. By introducing concepts such as "outer boundary," "primitive set," "Property A," "knot," "valid pair group,"…

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.…

We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.

We make an attempt at proving the Four Colour Theorem in six pages.

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…

I argue that there is no 4-chromatic planar graph with a joinable pair of color identical vertices, i.e., given a 4-chromatic planar graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring of…

In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.

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

This paper presents a short and simple proof of the Four-Color Theorem that can be utterly checkable by human mathematicians, without computer assistance. The new key idea that has allowed it and the global structure of the proof are…

I argue that, given vertices u and v in a 4-chromatic graph G, if the color of u equals the color of v in every 4-coloring of G then G has no planar supergraph where u and v are adjacent. This is equivalent to the Four Color Theorem.

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and G\"oring conjectured the following stronger statement to also hold: the vertices of every plane graph can be…

Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable.…

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…