On a problem of colouring the real plane

Filip Guldan

Mathematica Bohemica (1991)

  • Volume: 116, Issue: 3, page 309-318
  • ISSN: 0862-7959

Abstract

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What is the least number of colours which can be used to colour all points of the real Euclidean plane so that no two points which are unit distance apart have the same colour? This well known problem, open more than 25 years is studied in the paper. Some partial results and open subproblems are presented.

How to cite

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Guldan, Filip. "On a problem of colouring the real plane." Mathematica Bohemica 116.3 (1991): 309-318. <http://eudml.org/doc/29324>.

@article{Guldan1991,
abstract = {What is the least number of colours which can be used to colour all points of the real Euclidean plane so that no two points which are unit distance apart have the same colour? This well known problem, open more than 25 years is studied in the paper. Some partial results and open subproblems are presented.},
author = {Guldan, Filip},
journal = {Mathematica Bohemica},
keywords = {vertex colouring; infinity graph; decomposition of the real plane; vertex colouring; infinity graph; decomposition of the real plane},
language = {eng},
number = {3},
pages = {309-318},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a problem of colouring the real plane},
url = {http://eudml.org/doc/29324},
volume = {116},
year = {1991},
}

TY - JOUR
AU - Guldan, Filip
TI - On a problem of colouring the real plane
JO - Mathematica Bohemica
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 116
IS - 3
SP - 309
EP - 318
AB - What is the least number of colours which can be used to colour all points of the real Euclidean plane so that no two points which are unit distance apart have the same colour? This well known problem, open more than 25 years is studied in the paper. Some partial results and open subproblems are presented.
LA - eng
KW - vertex colouring; infinity graph; decomposition of the real plane; vertex colouring; infinity graph; decomposition of the real plane
UR - http://eudml.org/doc/29324
ER -

References

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  2. P. Erdös, Some unsolved problems, Publ. Math. Inst Hung. Acad. Sci. 6 (1961), 221-254. (1961) MR0177846
  3. P. Erdös F. Harary W. T. Tutte, 10.1112/S0025579300005222, Mathematika 12 (1965), 118-122. (1965) MR0188096DOI10.1112/S0025579300005222
  4. P. Erdös, 10.1007/BF02579174, Combinatorica 1 (1981), 25-42. (1981) MR0602413DOI10.1007/BF02579174
  5. P. Frankl, 10.1016/S0195-6698(80)80045-X, European J. Comb. I (1980), 101-106. (1980) Zbl0463.05043MR0587524DOI10.1016/S0195-6698(80)80045-X
  6. H. Hadwiger, Ungelöste Probleme No. 40, Elemente der Math. 16 (1961), 103-104. (1961) MR0133734
  7. H. Hadwiger H. Debrunner V. Klee, Combinatorial Geometry in the Plane, Holt, Reinehart and Winston, New York (1964). (1964) MR0164279
  8. D. G. Larman C. A. Rogers, 10.1112/S0025579300004903, Mathematika 19 (1972), 1-24. (1972) MR0319055DOI10.1112/S0025579300004903
  9. [9J L. Moser W. Moser, 10.1017/S1446788700014865, Canad. Math. Bull. 4 (1961), 187-189. (1961) DOI10.1017/S1446788700014865
  10. N. Wormald, A 4-chromatic graph with a special plane drawing, J. Austral. Math. Soc. Ser. A 28 (1979), 1-8. (1979) Zbl0418.05026MR0541161

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