12.7 Hamilton Cycles
3 min read•june 18, 2024
are a fascinating aspect of , showing how vertices connect in a single loop. They differ from , which focus on edges instead. Understanding these cycles helps us grasp complex network structures and solve real-world problems.
Complete graphs always have Hamilton cycles, and we can calculate how many using a simple formula. Finding the minimum weight Hamilton is crucial for optimization problems like the , which has practical applications in logistics and route planning.
Hamilton Cycles
Hamilton cycles vs Euler circuits
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- Hamilton cycles visit each in a graph exactly once, starting and ending at the same vertex without repeating any vertices except the starting/ending vertex
- Euler circuits visit each in a graph exactly once, starting and ending at the same vertex and may repeat vertices but not edges
- A graph can have both Hamilton cycles and Euler circuits ( ), only one, or neither
- Example: In a complete graph , a Hamilton is 1-2-3-4-1 and an Euler circuit is 1-2-3-1-4-2-4-3-1
Number of cycles in complete graphs
- Complete graphs have vertices with each vertex connected to every other vertex
- The number of Hamilton cycles in a complete graph is calculated using the formula
- represents the possible orderings of the remaining vertices after choosing a starting vertex
- Dividing by 2 eliminates duplicate cycles in opposite directions (1-2-3-4-1 and 1-4-3-2-1 are the same cycle)
- Example: A complete graph has Hamilton cycles
Minimum weight Hamilton cycles
- have numerical values (weights) assigned to their edges
- The of a Hamilton cycle is the sum of the weights of the edges in the cycle
- To find the Hamilton cycle with the minimum total weight:
- List all possible Hamilton cycles in the graph
- Calculate the total weight of each cycle by summing the weights of its edges
- Compare the total weights and identify the cycle(s) with the minimum value
- Example: In a weighted complete graph with weights 1-2: 4, 1-3: 2, 1-4: 6, 2-3: 3, 2-4: 5, 3-4: 1, the Hamilton cycles and their total weights are:
- 1-2-3-4-1: 4 + 3 + 1 + 6 = 14
- 1-2-4-3-1: 4 + 5 + 1 + 2 = 12
- 1-3-2-4-1: 2 + 3 + 5 + 6 = 16
- 1-3-4-2-1: 2 + 1 + 5 + 4 = 12
- 1-4-2-3-1: 6 + 5 + 3 + 2 = 16
- 1-4-3-2-1: 6 + 1 + 3 + 4 = 14
- The Hamilton cycles with the minimum total weight of 12 are 1-2-4-3-1 and 1-3-4-2-1
- This concept is closely related to the traveling salesman problem in optimization
Graph Theory Concepts Related to Hamilton Cycles
- Graph theory provides the foundation for understanding Hamilton cycles
- A in a graph is a sequence of vertices connected by edges, where no vertex is repeated
- A cycle is a path that starts and ends at the same vertex
- refers to the relationship between two vertices connected by an edge
- A is a graph that contains at least one Hamilton cycle
- The existence of a Hamilton cycle in a graph depends on various factors, including the graph's structure and connectivity
Key Terms to Review (20)
$K_4$: $K_4$ is a complete graph with four vertices, meaning that every pair of distinct vertices is connected by a unique edge. This graph serves as a fundamental structure in graph theory, particularly in the study of Hamilton cycles, as it provides a simple example of a graph where every vertex can be visited exactly once before returning to the starting vertex, thus forming a Hamiltonian circuit. Understanding $K_4$ helps in exploring properties like connectivity, graph traversal, and the criteria for Hamiltonicity in more complex graphs.
$K_n$: $K_n$ represents the complete graph on $n$ vertices, which means that every pair of distinct vertices is connected by a unique edge. It is a fundamental concept in graph theory, showcasing maximum connectivity and serving as a model for various real-world scenarios, including network design and social interactions. Understanding $K_n$ helps in exploring Hamiltonian cycles, where one seeks to find a cycle that visits each vertex exactly once and returns to the starting vertex.
Adjacency: Adjacency refers to the relationship between two vertices in a graph when they are directly connected by an edge. This concept is crucial in understanding the structure and properties of graphs, as it defines how vertices interact and relate to each other. Adjacency can impact various features of a graph, such as connectivity, pathfinding, and network flow, playing a significant role in more complex topics like cycles and graph comparisons.
Complete Graph: A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. This structure ensures that there are no missing connections, resulting in a highly interconnected network. The concept of complete graphs is crucial for understanding graph theory, as they serve as a benchmark for comparing the connectivity and efficiency of other graph structures.
Cycle: A cycle in graph theory is a path that starts and ends at the same vertex with no other vertices repeated. It forms a closed loop or circuit within the graph structure.
Cycle: A cycle in graph theory refers to a path that starts and ends at the same vertex, containing at least one edge, with no other repetitions of vertices and edges. This concept is crucial for understanding how graphs are structured and navigated, as cycles can affect various properties and operations within a graph, including traversal and connectivity.
Directed path: A directed path in a graph is a sequence of vertices where each adjacent pair is connected by a directed edge, following the direction from one vertex to the next. The order of vertices and the direction of edges are crucial for defining such a path.
Edge: An edge is a connection between two vertices in a graph. It can be directed (with a direction) or undirected (without a direction).
Edge: An edge is a fundamental component of a graph that represents a connection or relationship between two vertices (or nodes). Edges can be directed or undirected, and they can have weights associated with them, representing the cost or distance between the connected vertices. Understanding edges is crucial for analyzing graph structures, navigating graphs, and exploring various properties like circuits and paths.
Euler circuits: An Euler circuit is a path in a graph that visits every edge exactly once and returns to the starting vertex. This concept is important for understanding how to traverse networks efficiently, as it helps identify whether a graph can be traversed in such a way, based on the degrees of its vertices. Euler circuits connect to various real-world applications like optimizing routes and circuit design, highlighting their significance in graph theory.
Graph Theory: Graph theory is a branch of mathematics that studies the properties and relationships of graphs, which are mathematical structures used to model pairwise relationships between objects. It provides essential tools for analyzing various structures and processes, from social networks to transportation systems. Understanding graph theory helps in exploring complex connections, paths, and cycles within different contexts.
Hamilton circuits: A Hamilton circuit is a path in a graph that visits each vertex exactly once and returns to the starting vertex. It differs from an Euler circuit, which traverses each edge exactly once.
Hamilton cycles: Hamilton cycles are closed loops in a graph that visit each vertex exactly once before returning to the starting vertex. They are named after mathematician William Rowan Hamilton, who studied these types of paths in the context of graph theory. Understanding Hamilton cycles is essential for solving various problems in fields like computer science, operations research, and combinatorial optimization.
Hamiltonian graph: A Hamiltonian graph is a type of graph that contains a Hamiltonian cycle, which is a cycle that visits every vertex exactly once and returns to the starting vertex. This concept connects deeply with various properties of graphs, highlighting their structure and the potential for specific traversals. Understanding Hamiltonian graphs is essential for exploring more complex graph-related problems, especially when considering pathways and connectivity within networks.
Minimum weight Hamilton cycles: Minimum weight Hamilton cycles are special types of Hamiltonian cycles that not only visit each vertex in a graph exactly once but also have the lowest possible total weight or cost associated with the edges traversed. These cycles are particularly important in optimization problems, where the goal is to find the most efficient route that minimizes total distance or cost. Finding such cycles involves considering all possible Hamiltonian cycles and selecting the one with the minimum total weight, making them crucial in fields like logistics and network design.
Path: In the context of graph theory and related fields, a path is a sequence of edges that connects a sequence of vertices without revisiting any vertex. Paths are essential for understanding various structures and algorithms within graph representation, as they reveal how different points are interconnected. They play a crucial role in navigating graphs and analyzing routes in different scenarios.
Total weight: Total weight is the sum of the weights assigned to the edges in a weighted graph. It is used to evaluate the overall cost or distance of traversing all edges in a specific path or cycle.
Traveling salesman problem: The traveling salesman problem (TSP) is a classic optimization problem that focuses on finding the shortest possible route for a salesman to visit a set of cities and return to the origin city. It highlights the challenges of finding efficient paths in a weighted graph, where cities are represented as vertices and the distances between them as edges. Solving TSP is crucial in various applications, such as logistics, manufacturing, and scheduling.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.
Weighted graphs: A weighted graph is a type of graph in which each edge has an associated numerical value, known as a weight. These weights can represent various measures such as distance, cost, or time, making weighted graphs useful for modeling real-world scenarios where different paths have different costs. In the context of Hamilton cycles, weighted graphs help determine the most efficient route that visits each vertex exactly once and returns to the starting vertex, taking the weights into account.