Every planar map is four colorable

Appel, K. I.; Haken, Wolfgang

doi:10.1090/s0002-9904-1976-14122-5

Cited by 578 publications

(220 citation statements)

References 2 publications

(2 reference statements)

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“…There are of course, particular cases such as the solution of the celebrated four colour problem, which states that χ = 4 for any planar map [30]. χ = 2 trivially for a planar square lattice and a three colouring algorithm also exists as a consequence of the solution of the six vertex model [31].…”

Section: Higher State Potts Modelsmentioning

confidence: 99%

Ising spins on thin graphs

Baillie¹,

Johnston²,

Kownacki³

1994

Nuclear Physics B

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The Ising model on "thin" graphs (standard Feynman diagrams) displays several interesting properties. For ferromagnetic couplings there is a mean field phase transition at the corresponding Bethe lattice transition point. For antiferromagnetic couplings the replica trick gives some evidence for a spin glass phase. In this paper we investigate both the ferromagnetic and antiferromagnetic models with the aid of simulations. We confirm the Bethe lattice values of the critical points for the ferromagnetic model on φ 3 and φ 4 graphs and examine the putative spin glass phase in the antiferromagnetic model by looking at the overlap between replicas in a quenched ensemble of graphs. We also compare the Ising results with those for higher state Potts models and Ising models on "fat" graphs, such as those used in 2D gravity simulations.

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Section: Higher State Potts Modelsmentioning

confidence: 99%

Ising spins on thin graphs

Baillie¹,

Johnston²,

Kownacki³

1994

Nuclear Physics B

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“…Herein, we report that the global topology of the antiphase and chiral domain patterns in Fe x TaS 2 appears to be complex, but can be neatly understood in terms of the four color theorem [21][22][23] as well as a tensorial color theorem associated with two-step proper coloring. The four color theorem, which was empirically known to cartographers before the 17th century,…”

Section: ■ Introductionmentioning

confidence: 99%

Color Theorems, Chiral Domain Topology, and Magnetic Properties of Fe_xTaS₂

et al. 2014

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, superconductivity [6][7][8][9] , and discommensurations [10][11][12][13][14][15] . Intercalation of other transition metal ions between the MC 2 layers gives rise to distinct superstructures, leading to significant changes in crystallographic structures and physical properties. Fe-intercalated TaS 2 shows highly anisotropic ferromagnetism at low temperatures [16][17][18][19][20] . . Magnetic hysteresis curves of the crystals were obtained using a Quantum DesignMagnetic Property Measurement System, and the real Fe compositions were estimated from the saturation magnetic moments in the magnetic hysteresis curves with the assumption that each Supplementary Information, section S1). The distinct feature between the 2a×2a and √3a×√3a superstructures in Fe x TaS 2 is the different stacking sequence of the 2D supercells along the c-axis. Specifically, the 2a×2a superstructure consists of identically stacked 2D supercells (i.e., AA-type stacking), while the √3a×√3a superstructure contains shifted 2D supercells with AB-type stacking, as shown in Fig. 1b and Fig. 1f, -5 -respectively. These different stacking sequences result in the centrosymmetric P6 3 /mmc and noncentrosymmetric and chiral P6 3 22 space groups for the 2a×2a and √3a×√3a superstructures, respectively.We found complicated configurations of antiphase domains in the dark-field images of There is an extinction rule for the dark-field images of antiphase boundaries in the 2a×2a superstructure. For example, the antiphase boundary between the BB-type and CC-type antiphase domains appears in the S1=( /2 00) (Fig 2a) and S2=(0 1/2 0) (Fig. 2b) dark-field images, but disappears in the S3=(1/2 /2 0) dark-field image of Fig. 2c (see also Supplementary Information, section S3). Each antiphase boundary becomes invisible in a dark-field image taken using one out of three superlattice spots (namely, S1, S2, or S3), when no antiphase shift at the boundary exists along a certain superlattice modulation wave vector. This absence of antiphase shifts at the antiphase boundary leads to the extinction rule for the antiphase boundaries in superlattice dark-field images. This rule is summarized in Fig. 3, showing the local structures near boundaries between two antiphase domains. The boundaries are highlighted with yellow bands, and the three directions of the superlattice modulation wave vectors are denoted by S1, S2, and S3, respectively, as shown in Fig. 1a. The red, yellow, blue, and green circles correspond to -6 -AA-, BB-, CC-, and DD-type superstructures, respectively, which are associated with four possible origins of the 2a×2a Fe superstructure. It is evident that the superlattice modulation along only one out of three equivalent crystallographic directions does not show any antiphase shift; this is indicated by light green dashed lines (along the S1 direction), light blue dashed lines (along the S2 direction), and pink dashed lines (along the S3 direction). For example, the antiphase boundary between BB-type and CC-type (or AA-type and DD-type) antiphase domains ha...

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“…Secondly, this result is not a consequence of the fourcolourability of planar graphs first conjectured by Francis Guthrie and proved by Appel and Haken (1977). Consider a pedigree in which three males each mate with each of three females.…”

Section: Notesmentioning

confidence: 82%