Question

Find a Hamiltonian cycle in $Q_3$.

Short answer

The Hamiltonian cycle in $Q_3$ is: 000 -> 001 -> 011 -> 010 -> 110 -> 111 -> 101 -> 100 -> 000.

Step-by-step solution

Understanding the Graph

First, understand that $Q_3$ is the 3-dimensional hypercube. It consists of 8 vertices (each vertex represents a 3-bit binary number) and 12 edges. Vertices are connected if their binary numbers differ by exactly one bit.

Listing the Vertices

The vertices of $Q_3$ are $000,001,010,011,100,101,110,111$. Each represents a corner of the cube.

Choosing a Starting Vertex

Start from any vertex, say $000$. A Hamiltonian cycle will visit each vertex exactly once and return to the starting vertex.

Construct the Cycle

From $000$, move to the adjacent vertex $001$, then $011$, $010$, $110$, $111$, $101$, $100$, and finally back to $000$.

Verification

Check that each vertex is visited exactly once and connections between each step follow the hypercube's edges. The cycle $000\to 001\to 011\to 010\to 110\to 111\to 101\to 100\to 000$ forms a valid Hamiltonian cycle in $Q_3$.

Key concepts

Hypercube Graph

A hypercube graph is a fascinating and complex structure that generalizes the concept of a cube to higher dimensions. In graph theory, a hypercube of dimension $n$, usually denoted as $Q_n$, is constructed by:

• Using vertices that represent all possible $n$-bit binary numbers.
• Connecting two vertices with an edge if their binary numbers differ by exactly one bit.

For example, the 3-dimensional hypercube, $Q_3$, has:

• 8 vertices, since you can form 8 different 3-bit numbers (from 000 to 111).
• 12 edges, where each vertex connects to 3 others, reflecting the dimensionality.

The structure of a hypercube graph makes it highly symmetrical and useful for visualizing problems in higher dimensions. It's widely used in various applications like computer science and combinatorics, where it models many-to-many connections in parallel computing systems.

Hamiltonian Graph

A Hamiltonian graph is a type of graph that contains a Hamiltonian cycle. This is a cycle that visits every vertex exactly once, and returns to the starting point. Finding such a cycle in certain types of graphs can be crucial for solving complex optimization problems.Characteristics of Hamiltonian graphs include:

• Completeness: While not all graphs are Hamiltonian, complete graphs always contain a Hamiltonian cycle.
• Cycle Construction: Requires ensuring every vertex is visited once without repetition, except for the starting and ending vertex.

Hamiltonian cycles are particularly interesting in theory due to their potential to solve problems like the Traveling Salesman Problem. In practice, determining a Hamiltonian cycle can involve creative and strategic pathfinding, which makes them an enthralling concept in graph theory. In the case of $Q_3$, as demonstrated in the exercise, one such cycle is: $000\to 001\to 011\to 010\to 110\to 111\to 101\to 100\to 000$. This path confirms each vertex is visited once, looping back to the origin.

Graph Theory

Graph theory is a field of mathematics centered around the study of graphs, which consist of vertices (or nodes) connected by edges. It's a versatile framework with applications in computer networks, social sciences, biology, and more. Some fundamental concepts in graph theory include:

• Vertices and Edges: The basic building blocks of graphs, representing points and connections respectively.
• Paths and Cycles: Paths refer to sequences of edges linking vertices, while cycles return to the starting vertex.

Graphs can be further categorized based on properties such as direction (directed vs. undirected), weight on edges, and connectivity. Advanced concepts in graph theory include:

• Planarity: Determining if a graph can be redrawn in a plane without edge crossings.
• Graph Coloring: Assigning colors to vertices so adjacent ones are distinctly colored, used in scheduling and resource allocation.

Graph theory not only aids in mathematical analysis but also provides a robust language for describing and solving real-world problems. In the context of the hypercube and Hamiltonian graphs, the ability to navigate these graphs efficiently can translate into improved algorithms and computing strategies.