Topology coloring

Authors
hmaid erfanianoraei dehrokhi
majid erfanian oraei
Abstract
Introduction

The aim of this study, is painting of topological surfaces with the least number of colors without the distance, and the colors have a border. For this purpose, we need a color mapping. In this mapping, we have not any fixed point, and we can colorable the map with least colors.

Definition: Let f∶ X → X be a graph without a fixed point. f is colorable with k colors, if there is C={C_1,…,C_K}, where all C_i do not include {(x, f(x)}. Or similarly, for every i=1,…, k, there is the equation C_i ∩ f(C_i )=∅.

Also, we define some concepts such as Compression, Metric, or non-Compression of space. Also, to achieve the desired result of each space, we change the properties of the maps.

Material and methods

In this work, first, we define the properties and conditions of the color mapping and color number. Also, by the study of properties of each space, we choose the best of space. One of the best conditions of this space is the lowest color number and higher efficiency. Finally, we proved that this number is finite, and we can do coloring space with some maps and conversely.

Results and discussion

In this work, we define the properties and conditions of the color mapping and color number. We presented some theorems and Lemma in the article and proved them for coloring of any space by coloring map, the coloring number is at least 3 and at most is a n+3. Also, we proved the coloring number finite and we can do coloring space with some maps and conversely.

Conclusion

The following conclusions were drawn from this research.




the coloring number is at least 3.
the coloring number is at most n+3.
coloring number is finite and we can do coloring space with some maps.
We can do the coloring of any space by the finite coloring map.
Keywords
Color mappings
color numbers
dot coverage
Continuity
Homogeneity
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Mathematical Research
Volume 8, Issue 2 - Serial Number 21
Spring 2022
Pages 138-151