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A Geometric and Visual Perspective on the Four Color Map Theorem and K5 Non-Planarity and Their Connection
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Submitted:
16 January 2024
Posted:
17 January 2024
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Abstract
This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.
Keywords:
Subject: Computer Science and Mathematics - Discrete Mathematics and Combinatorics
1. One Vertex N1 or K1:
If we represent it as circle:
2. Two Vertex isolated N2:
If we represent it as circle:
2.1. Two vertices adjacent to each other K2:
If we represent it as circle:
3.1. Three vertices adjacent to each other K3:
If we represent it as circle:
3.2. Also another variety of 3 vertices where only middle vertex connect two other two vertices:
If we represent it as circle:
4.1. Four vertices in a simple rectangular arrangement:
If we represent it as circle:
4.2. Four vertices in K4 rectangular arrangement:
If we represent it as circle:
OR
Background:
The Four Color Theorem has been a fundamental problem in graph theory for over a century. It posits that any planar graph can be colored with no more than four colors such that no two adjacent regions share the same color.
Abstract:
This paper presents a unique approach to the Four Color and K5 non planarity and draws a conclusion that both are basically the same problem and their connection goes to geometry of interaction properties of a group of circles(Kissing number problem) . Using a geometric representation of planar graphs with circles and tangents. We explore the implications of the kissing number problem in two dimensions to provide a new perspective on the theorem.
The Four Color Theorem in this Context:
The theorem states that no more than four colors (or planes, in your interpretation) are needed to ensure that no two adjacent circles (vertices) share the same color (or are on the same plane).
Lemma: Kissing Number in 2D Plane
In a 2D plane, the maximum number of equal-sized circles that can touch another circle without any of them overlapping is 4. The four color map theorem is deeply connected with this geometrical nature of interaction of circles or any closed graphs in a 2d plane.
Proof by Contradiction(Four color map theorem):
Assume that a planar graph needs at least five colors. This would mean, in your circle model, that there exists a set of five circles where each circle must touch all the others.
However, as per the kissing number problem in 2D, a maximum of four circles can mutually touch one another.
This contradiction would suggest that the initial assumption (needing at least five colors) is false, hence supporting the Four Color Theorem.
Proof by Contradiction (K5 non planarity):
Assume that a planar graph needs at least five colors. This would mean, in your circle model, that there exists a set of five circles where each circle must touch all the others. However, as per the kissing number problem in 2D, a maximum of four circles can mutually touch one another. This contradiction would suggest that the initial assumption (needing at least five colors) is false, hence supporting the non-planarity of K5.
Connection Between K5 and Four Color Map Theorem:
It's worth noting that the non-planarity of K5 and the Four Color Map Theorem are deeply connected. Both problems revolve around the concept of coloring, and the kissing number
problem in 2D helps illustrate this connection. While the Four Color Theorem traditionally applies to the coloring of regions (faces) formed by the edges of a planar graph, the non- planarity of K5 demonstrates the limitations of such colorings, reinforcing the importance of the Four Color Theorem.
Closing and Conclusion:
In this paper, we have explored a fascinating intersection between geometry and graph theory, shedding light on the intriguing concept of the "kissing number" of circles in a 2D plane. We began by introducing the fundamental idea of representing planar graphs using groups of circles, where circles symbolize vertices, and tangents between them represent edges. This visual representation not only simplifies complex graph structures but also offers new insights into long-standing problems.
Our journey led us to the heart of the "kissing number" problem, a classic question in mathematics. We established the essential theorem that in a 2D plane, the maximum number of circles that can touch another circle without overlapping is 4. This simple yet profound result has far-reaching implications in various fields, from geometry to network design.
By providing a lemma and proof, we have contributed to the comprehensive understanding of this intriguing problem. We have clarified that the size of the circles need not be uniform; it is their relative positions and non-overlapping nature that define the kissing
References
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