Related papers: Combinatorial Stacks and the Four-Colour Theorem

A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other…

We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $\tau$ is parametrized by…

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be…

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…

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…

To each permutation $\sigma$ in $S_{n}$ we associate a triangulation of a fixed $(n+2)$-gon. We then determine the fibers of this association and show that they coincide with the sylvester classes depicted By Novelli, Hivert and Thibon. A…

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 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,…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian…

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).…

We introduce a new cohomology theory for planar trivalent graphs with perfect matchings. The graded Euler characteristic of the cohomology is a one variable polynomial called the 2-factor polynomial that, if nonzero when evaluated at one,…

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…

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring…

While planar graphs are flat from a topological viewpoint, we observe that they are not from a geometric one. We prove that every planar graph can be embedded into a surface consisting of spheres, glued together in a tree-like fashion. As a…

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…

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…

We present a ``reasonable'' statement about Lie algebras that is equivalent to the Four Color Theorem. The notions appearing in the statement also appear in the theory of finite-type invariants of knots (Vassiliev invariants) and…

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…