A Six Color Problem

Franklin, Philip

doi:10.1002/sapm1934131363

Cited by 71 publications

(53 citation statements)

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“…Heawood Although the Klein bottle has characteristic 0, Franklin (1934) proved that it is only 6-chromatic, and found the following &chromatic map on the Klein bottle: It has been shown that, with the sole exception of the Klein bottle, equality holds in (111.2) for every surface S of characteristic n C2. This result is known as the map colour theorem (see Ringel, 1974 …”

Section: The Platonic Graphsmentioning

confidence: 99%

Graph Theory with Applications

Bondy¹

1976

6,948

3,571

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Section: The Platonic Graphsmentioning

confidence: 99%

Graph Theory with Applications

Bondy¹

1976

6,948

3,571

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“…Thus, as is well known, ju(2-sphere) = 4, «(projective plane) = 6, jtt(torus) = 7 and jn(Klein's bottle) = 8 (see [5]). The following lemma is obvious:…”

Section: Corollarymentioning

confidence: 71%

“…G(ty) contains the subgroups G" G2, G3 where, in addition to the identity, G, contains the permutations (1, 3)(5, 7), (1, 7)(3, 5) and (1,5) (3, 7); G2 contains (2, 6)(4, 8), (2,8)(4, 6) and (2, 4) (6,8); and G3 contains (1, 2)(3, 4)(5, 6)(7, 8). G, and G2 are Klein groups.…”

Section: -37 46 -35 68 -57mentioning

confidence: 99%

3-pseudomanifolds with preassigned links

Altshuler¹

1978

Trans. Amer. Math. Soc.

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Abstract. A 3-pseudomanifold is a finite connected simplicial 3-complex % such that every triangle in % belongs to precisely two 3-simplices of %, the link of every edge in % is a circuit, and the link of every vertex in % is a closed 2-manifold. It is proved that for every finite set 2 of closed 2-manifolds, there exists a 3-pseudomanifold % such that the link of every vertex in % is homeomorphic to some S e 2, and every S & 2 is homeomorphic to the link of some vertex in %.

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“…Anyway, these experimental results are in agreement with graph col-oring theory. As a matter of fact, that theory also states that at most 6 colors are required to color any map on the plane or the sphere [8]. Of course, and as repeatedly mentioned in the paper, the Four Color Theorem [17] demonstrates that the necessary number of colors is 4.…”

Section: Theory and Parameters In Practicementioning

confidence: 87%

Efficient Loopy Belief Propagation Using the Four Color Theorem

Timofte

Gool

2013

Advanced Topics in Computer Vision

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Summary. Recent work on early vision such as image segmentation, image denoising, stereo matching, and optical flow uses Markov Random Fields. Although this formulation yields an NP-hard energy minimization problem, good heuristics have been developed based on graph cuts and belief propagation. Nevertheless both approaches still require tens of seconds to solve stereo problems on recent PCs. Such running times are impractical for optical flow and many image segmentation and denoising problems and we review recent techniques for speeding them up. Moreover, we show how to reduce the computational complexity of belief propagation by applying the Four Color Theorem to limit the maximum number of labels in the underlying image segmentation to at most four. We show that this provides substantial speed improvements for large inputs, and this for a variety of vision problems, while maintaining competitive result quality.

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A Six Color Problem

Cited by 71 publications

References 0 publications

Graph Theory with Applications

Graph Theory with Applications

3-pseudomanifolds with preassigned links

Efficient Loopy Belief Propagation Using the Four Color Theorem

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