| Title:
|
On a problem of colouring the real plane (English) |
| Author:
|
Guldan, Filip |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
116 |
| Issue:
|
3 |
| Year:
|
1991 |
| Pages:
|
309-318 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
vertex colouring |
| Keyword:
|
infinity graph |
| Keyword:
|
decomposition of the real plane |
| MSC:
|
05C15 |
| idZBL:
|
Zbl 0758.05052 |
| idMR:
|
MR1126452 |
| DOI:
|
10.21136/MB.1991.126170 |
| . |
| Date available:
|
2009-09-24T20:46:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126170 |
| . |
| Reference:
|
[1] R. B. Eggleton P. Erdös D. K. Skilton: Colouring the real line.J. Comb. Theory Ser. B 39 (1985), 86-100. MR 0805458, 10.1016/0095-8956(85)90039-5 |
| Reference:
|
[2] P. Erdös: Some unsolved problems.Publ. Math. Inst Hung. Acad. Sci. 6 (1961), 221-254. MR 0177846 |
| Reference:
|
[3] P. Erdös F. Harary W. T. Tutte: On dimension of a graph.Mathematika 12 (1965), 118-122. MR 0188096, 10.1112/S0025579300005222 |
| Reference:
|
[4] P. Erdös: On combinatorial problems which I would most like to see solved.Combinatorica 1 (1981), 25-42. MR 0602413, 10.1007/BF02579174 |
| Reference:
|
[5] P. Frankl: Extremal problems and coverings of the space.European J. Comb. I (1980), 101-106. Zbl 0463.05043, MR 0587524, 10.1016/S0195-6698(80)80045-X |
| Reference:
|
[6] H. Hadwiger: Ungelöste Probleme No. 40.Elemente der Math. 16 (1961), 103-104. MR 0133734 |
| Reference:
|
[7] H. Hadwiger H. Debrunner V. Klee: Combinatorial Geometry in the Plane.Holt, Reinehart and Winston, New York (1964). MR 0164279 |
| Reference:
|
[8] D. G. Larman C. A. Rogers: The realization of distances within sets in Euclidean space.Mathematika 19 (1972), 1-24. MR 0319055, 10.1112/S0025579300004903 |
| Reference:
|
[9J L. Moser W. Moser: Solution to Problem 10.Canad. Math. Bull. 4 (1961), 187-189. 10.1017/S1446788700014865 |
| Reference:
|
[10] N. Wormald: A 4-chromatic graph with a special plane drawing.J. Austral. Math. Soc. Ser. A 28 (1979), 1-8. Zbl 0418.05026, MR 0541161 |
| . |
