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### A new proof of Grünbaum's 3 color theorem

O.V. Borodin
1997 Discrete Mathematics
A simple proof of Grfinbaum's theorem on the 3-colourability of planar graphs having at most three 3-cycles is given, which does not employ the colouring extension.  ...  In 1958, Gr6tzsch I-5] proved that every planar graph without cycles of length three is 3-colourable. In 1963, Griinbaum [6] extended this result as follows: Theorem 1.  ...  As a result, the main Theorem 1 was also deprived of its proof. In 1974, Aksenov [1] proved the following version of Claim 2. Theorem 3. Let G be a plane graph with faces of lengths 3, 4, and 5.  ...

### Grünbaum coloring and its generalization to arbitrary dimension [article]

S. Lawrencenko, M.N. Vyalyi, L.V. Zgonnik
2016 arXiv   pre-print
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 striking significances of Gr\"unbaum's Conjecture in the 2-dimensional case is its equivalence to the 4-Color Theorem.  ...  the First Pacific Rim Conference on Mathematics (January [19] [20] [21] [22] [23] 1998, City University of Hong Kong), on the subject of the 3-sphere recognition problem.  ...

### Arrangements of Pseudocircles: Triangles and Drawings [chapter]

Stefan Felsner, Manfred Scheucher
2018 Lecture Notes in Computer Science
Theorem 8. Every intersecting arrangement A of pseudocircles fulfills p 3 (A) ≤ 2 3 n 2 + O(n). Proof. Let A be an intersecting arrangement of n ≥ 4 pseudocircles.  ...  Hence, there are p 3 (A) + p 3 (B) + δ − τ − 1 triangles in C. Proof of Theorem 1(iii). We use A 12 , the arrangement shown in Figure 3 (a), in the role of A for our recursive construction.  ...

### Arrangements of Pseudocircles: Triangles and Drawings [article]

Stefan Felsner, Manfred Scheucher
2020 arXiv   pre-print
For pairwise intersecting arrangements with digons we have a lower bound of p_3 ≥ 2n/3, and conjecture that p_3 ≥ n-1.  ...  We expect that the lower bound p_3(A) ≥ 4n/3 is tight for infinitely many simple arrangements.  ...  Hence, there are p 3 (A) + p 3 (B) + δ − τ − 1 triangles in C. Proof of Theorem 1(iii). We use A 12 , the arrangement shown in Figure 3(a) , in the role of A for our recursive construction.  ...

### Arrangements of Pseudocircles: Triangles and Drawings

Stefan Felsner, Manfred Scheucher
2020 Discrete & Computational Geometry
For pairwise intersecting arrangements with digons we have a lower bound of p 3 ≥ 2 n / 3 , and conjecture that p 3 ≥ n - 1 .  ...  With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with p 3 ( A ) / n → 16 / 11 = 1 . 45 ¯ .  ...  Hence, there are p 3 (A ) + p 3 (B) + δ − τ − 1 triangles in C . Proof of Theorem 2.1(iii) We use A 12 , the arrangement shown in Fig. 3 (a) , in the role of A for our recursive construction.  ...

### On Codimension one Embedding of Simplicial Complexes [article]

Anders Björner, Afshin Goodarzi
2017 arXiv   pre-print
The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.  ...  It is shown that such a complex must satisfy a certain homological condition.  ...  Part of the research of the second author has been made possible by the grant KAW-stipendiet 2015.0360 from the Knut and Alice Wallenberg's Fondation.  ...

### Grünbaum colorings of toroidal triangulations

Michael O. Albertson, Hannah Alpert, sarah-marie belcastro, Ruth Haas
2010 Journal of Graph Theory
We prove that if G is a triangulation of the torus and χ(G) = 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.  ...  working on open problems, some of whose solutions appear here.  ...  Thanks to sarah-marie belcastro's topological graph theory class at the 2007 session of the Hampshire College Summer Studies in Mathematics (particularly Miles Edwards and Nate Harman), for enthusiastically  ...

### Grünbaum Colorings of Toroidal Triangulations [article]

Michael O. Albertson, Hannah Alpert, sarah-marie belcastro, Ruth Haas
2008 arXiv   pre-print
We prove that if G is a triangulation of the torus and \chi(G) \neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.  ...  working on open problems, some of whose solutions appear here.  ...  Thanks to sarah-marie belcastro's topological graph theory class at the 2007 session of the Hampshire College Summer Studies in Mathematics (particularly Miles Edwards and Nate Harman), for enthusiastically  ...

### Relating embedding and coloring properties of snarks

Bojan Mohar, Eckhard Steffen, Andrej Vodopivec
2008 Ars Mathematica Contemporanea
We then relate the defect with the resistance r(G) of a cubic graph G which is the size of a minimum color class of a 4-edge-coloring of G.  ...  To describe the deviation from polyhedrality, we define the defect of a graph and use it to study embeddings of superpositions of cubic graphs into orientable surfaces.  ...  Theorem 12. If there exists a snark G embeddable into a surface S with d S (G) < r(G) 2 then there exists a snark G embeddable into S with d S (G ) = 0. Proof. We follow the proof of Theorem 10.  ...

### On regulating sets and the disparity of planar cubic graphs

Herbert Fleischner
By Grunbaum's Theorem [2, Theorem 12.8], D F has a 3-point-coloring C D with the color classes (1'), (2'), (3').  ...  If (The proof of Corollary 2 is immediate from the proof of Theorem 5.) From Corollary 2 and Theorem 4 the following corollary is immediate. COROLLARY 3.  ...

### Acyclic list 7-coloring of planar graphs

O. V. Borodin, D. G. Fon-Der Flaass, A. V. Kostochka, A. Raspaud, E. Sopena
2002 Journal of Graph Theory
The acyclic list chromatic number of every planar graph is proved to be at most 7. ß 2002 Wiley Periodicals, Inc. J Graph Theory 40: [83][84][85][86][87][88][89][90] 2002  ...  This completes the proof of Theorem 1. ( 2 ) 2 Take a new vertex in each face of the 3-dimensional cube and join it to all the middle points of its edges, considering them as new vertices.  ...  7-vertices, so that ch Ã ðvÞ ! À2 þ 8 Â 1 4 ¼ 0. This completes the proof of Theorem 2.  ...

### Signed Diagonal Flips and the Four Color Theorem

Shalom Eliahou
1999 European journal of combinatorics (Print)
Finally, we show that this conjecture, if true, would imply the four color theorem.  ...  We introduce a signed version of the diagonal flip operation.  ...  Kauffman for pointing out to me the existence of Kryuchkov's preprint in September 1998, and for sending me a copy of it.  ...

### Acyclic Choosability of Graphs with Small Maximum Degree [chapter]

Daniel Gonçalves, Mickaël Montassier
2005 Lecture Notes in Computer Science
The proof of (b) uses a backtracking greedy algorithm and Burstein's theorem. ⋆ goncalve@labri.fr ⋆⋆ montassi@labri.fr  ...  A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle.  ...  In Section 3, we prove Theorem 8. Proof of Theorem 7 Let H be a counterexample to Theorem 7 with minimum order.  ...

### On the acyclic choosability of graphs

Mickaël Montassier, Pascal Ochem, André Raspaud
2006 Journal of Graph Theory
Conjecture 1 is very strong, since it implies the celebrated result of Borodin (Theorem 1), and we know that its proof is tough.  ...  A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle.  ...  In 1979, Borodin proved Grünbaum's conjecture : Theorem 1 [Bor79] Every planar graph is acyclically 5-colorable.  ...

### Fair division and generalizations of Sperner- and KKM-type results [article]

Megumi Asada, Florian Frick, Vivek Pisharody, Maxwell Polevy, David Stoner, Ling Hei Tsang, Zoe Wellner
2017 arXiv   pre-print
Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds.  ...  For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma.  ...  We state this stronger version of Grünbaum's theorem below and briefly outline the proof; for details refer to [17] .  ...
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