Question

Prove that \(Q_{n}\) is bipartite for \(n=2,3,4, \ldots\)

Short answer

Hypercube graphs \(Q_n\) are bipartite for \(n \geq 2\) using an inductive argument.

Step-by-step solution

Understanding the Problem

To prove that the n-dimensional hypercube graph, denoted as \(Q_n\), is bipartite, we need to show that its vertices can be divided into two disjoint sets where there are no edges between vertices within the same set.

Base Case: Q2 is Bipartite

\(Q_2\) is a simple square, so we can label opposite corners with the same label and alternate, proving \(Q_2\) is bipartite. We assign two vertices as one set \(\{00,11\}\) and the other as \(\{01,10\}\), which meets the conditions for a bipartite graph.

Inductive Hypothesis

Assume that \(Q_k\) is bipartite for some \(k \geq 2\). This means we can partition the vertex set of \(Q_k\) into two sets, \(A_k\) and \(B_k\), such that no two vertices in the same set are adjacent.

Constructing Q(k+1)

The \( (k+1) \)-dimensional hypercube, \(Q_{k+1}\), can be constructed by taking two copies of \(Q_k\) and connecting corresponding vertices from the two copies.

Partitioning Q(k+1)

For \(Q_{k+1}\), assign one copy of \(Q_k\) with the set \(A_k\) labeled \(0\) in front and the other with \(B_k\) labeled \(1\) in front. Thus, if a vertex is in \((v, 0) \in Q_{k+1}\), it is in set \(A_k\), and if it is \((v, 1)\), it is in set \(B_k\). As a result, this maintains the bipartite nature of \(Q_{k+1}\) without intra-set edges.

Concluding the Induction

Since \(Q_k\) is bipartite and \(Q_{k+1}\) can be partitioned as described, \(Q_{k+1}\) remains bipartite. Thus, by the principle of mathematical induction, \(Q_n\) is bipartite for all \(n \geq 2\).

Key concepts

Hypercube Graphs

Hypercube graphs, often denoted as \(Q_n\), are fascinating objects in graph theory. These graphs represent an \(n\)-dimensional cube, with vertices corresponding to binary strings of length \(n\). Each vertex is labeled by a unique sequence of \(n\) bits (0s and 1s). Adjacency between vertices is defined by their bitwise difference. If two vertices differ by exactly one bit, they are connected by an edge. This unique property of hypercube graphs makes them a popular area of study in mathematics and computer science.

• For example, in a 3-dimensional hypercube \(Q_3\), there are 8 vertices, ranging from \(000\) to \(111\).
• The adjacency can be visualized as 3D geometric cubes.

Hypercube graphs are known for their frequent appearances in areas such as parallel computing and coding theory due to their structural regularity and ease of traversal.

Inductive Proof

Inductive proof is a powerful method used in mathematics to prove the truth of an infinite sequence of statements. It consists of two key steps: the base case and the inductive step. In proving properties of hypercube graphs, induction allows us to extend known properties from lower dimensions to higher ones.

• Base Case: Establish that the property holds for the initial value, often \(n = 2\).
• Inductive Step: Assuming the property holds for \(n = k\), prove it for \(n = k+1\).

By confirming these steps, one can conclude that the property in question holds for all subsequent integers. In our hypercube exercise, induction shows that if \(Q_k\) is bipartite, then so is \(Q_{k+1}\). This structured approach allows for a clean and systematic way to tackle complex graph properties and many other mathematical challenges.

Graph Theory

Graph theory is a branch of mathematics that studies points, or "vertices," connected by lines, or "edges." It provides a comprehensive framework to model relationships and interactions, widely applicable in networks, computer science, biology, and more. Within this theory, several specialized classes of graphs, such as bipartite graphs and hypercubes, offer distinct properties and applications.

• Bipartite Graphs: These are graphs whose vertices can be divided into two disjoint and independent sets, such that every edge connects a vertex in one set to a vertex in the other. A classic example is a chessboard, where pieces only move between different colors.
• Hypercube Graphs: Represents higher-dimensional cubes and serves as a typical model in parallel computing.

Graph theory's versatility and depth make it a central area of study for problems involving networks and connectivity, including verifying whether a hypercube graph is bipartite.

Mathematical Induction

Mathematical induction is a fundamental proof technique used to prove statements about all natural numbers. It involves showing two things: first, that a statement holds for an initial case (often \(n=1\) or \(n=2\)), and second, that if the statement holds for an arbitrary case \(n=k\), it also holds for \(n=k+1\).
The process can be illustrated as a domino effect: if you can ensure the first domino falls, and that each domino impacts the next, then all will eventually fall. In the context of our exercise, mathematical induction was used to show that hypercube graphs are bipartite for any dimension \(n\):

• The bipartite nature of \(Q_2\) serves as our base case.
• The inductive step assumes that \(Q_k\) is bipartite, and proves that \(Q_{k+1}\) is as well.

This "chain reaction" method is pivotal in mathematics, enabling the derivation of infinite truths from finite evidence.