Question

Show that an infinite graph \(G\) can be drawn in Euclidean 3-space if \(V(G)\) and \(E(G)\) can each be put in one-one correspondence with a subset of the set of real numbers.

Short answer

An infinite graph can be illustrated in Euclidean 3-dimensional space by mapping each vertex and edge bijectively onto a unique point in the space. This makes it possible to draw the graph without overlaps, thus allowing the drawing of an infinite graph.

Step-by-step solution

Define the graph and mappings

Define an infinite, graph \(G\) with the vertex set \(V(G)\) and edge set \(E(G)\). Assume mappings \(f: V(G) \rightarrow \mathbb{R}\) and \(g: E(G) \rightarrow \mathbb{R}\), such that both are one-to-one correspondences, meaning they are bijective mappings.

Consider the Euclidean Space

Euclidean 3-space is a three-dimensional space that includes every possible coordinate of form \((x, y, z)\) where \(x, y, z \in \mathbb{R}\). Since \(V(G)\) and \(E(G)\) are each mapped bijectively to a subset of real numbers, we can map all vertices and edges of \(G\) to distinct coordinates in Euclidean 3-space.

Explain the process

Every vertex \(v \in V(G)\) is mapped to a number \(f(v)\) and positioned at the coordinate \((f(v), 0, 0)\) in Euclidean 3-space. For every edge \(e \in E(G)\), connecting vertices \(v_i, v_j\), we map it to a number \(g(e)\) and draw a straight line between \((f(v_i), 0, 0)\) and \((f(v_j), 0, 0)\) through the point \((g(e), g(e), g(e))\). This creates a unique line in the 3D space as there cannot be two edges with the same mapping.

Conclude

Since we are leveraging three dimensions and both vertices and edges have unique coordinates, we are not limited by overlaps. We can thus construct a representation of the graph in the Euclidean 3-space, no matter the size of the graph. Based on this reasoning, we have shown that an infinite graph \(G\) can be drawn in Euclidean 3-space given \(V(G)\) and \(E(G)\) can each be put in one-one correspondence with a subset of the set of real numbers.

Key concepts

Bijective Mapping

In the context of the given exercise, understanding a bijective mapping is crucial to grasp how an infinite graph can be drawn in a three-dimensional space. A bijective mapping, also known as a one-to-one correspondence, is a type of function that relates every element of one set to a unique element of another set, with no two elements of the first set being related to the same element of the second set, and vice versa.

For instance, if we have a set of students and a set of student ID numbers, a bijective mapping would assign each student one unique ID number, and each ID number is associated with only one student. In graph theory, this concept enables us to associate every vertex and edge of an infinite graph with a unique real number. This is vital because it ensures that when we draw the graph in Euclidean 3-space, each vertex and edge occupies a distinct position, and there is no ambiguity or overlap.

Euclidean 3-Space

Euclidean 3-space is a mathematical concept that represents a three-dimensional space in which any point can be defined by three coordinates \( (x, y, z) \) where \( x, y, z \) are real numbers. It is essentially the three-dimensional extension of the Euclidean plane most students are familiar with from geometry.

This space is fundamental when attempting to visualize complex objects such as infinite graphs. The inherent property of Euclidean 3-space that makes it particularly useful for this exercise is its capacity to accommodate an infinite number of points without overlap, which is impossible in two dimensions. To place an infinite graph within this space is to assign to each vertex and edge a location that is described by three coordinates that are derived from the bijective mappings mentioned earlier.

Graph Theory

Graph theory is a field of mathematics that deals with problems related to graphs, which are mathematical structures used to model pairwise relations between objects. A graph consists of vertices (or nodes) and edges (lines that connect vertices). Graph theory provides a way of representing relationships in a visual and mathematical way, which is applicable to numerous fields, from computer science to biology.

When working with infinite graphs, the challenge is to represent the graph in a way that we can comprehend and analyze these relationships without running into the limitations of space. The step-by-step solution for our textbook exercise navigates these issues using bijective mappings and the properties of Euclidean 3-space, showcasing graph theory's ability to extend its application beyond simple, finite structures to encompass even the most complex of infinite systems.