11.2 Eulerian and Hamiltonian paths and cycles
3 min read•july 30, 2024
Eulerian and Hamiltonian paths and cycles are key concepts in graph theory. They help us understand how we can traverse graphs by visiting edges or vertices. These ideas have real-world applications in network design, route planning, and even DNA sequencing.
While Eulerian paths focus on visiting every edge once, Hamiltonian paths aim to visit every vertex once. Eulerian problems are generally easier to solve, but Hamiltonian problems are more complex. Understanding both helps us tackle various graph-related challenges in computer science and beyond.
Eulerian vs Hamiltonian Paths and Cycles
Definitions and Key Differences
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- visits every edge exactly once in a graph
- forms a closed path visiting every edge exactly once
- visits every vertex exactly once in a graph
- forms a closed path visiting every vertex exactly once
- Eulerian concepts focus on edges while Hamiltonian concepts focus on vertices
- Eulerian paths/cycles named after Leonhard Euler who solved problem (1736)
- Hamiltonian paths/cycles named after William Rowan Hamilton who invented (1857)
Complexity and Applications
- Eulerian paths/cycles depend on vertex degrees making them easier to characterize
- Hamiltonian paths/cycles depend on complex structural properties making them NP-complete problems
- Finding Eulerian paths/cycles generally simpler than Hamiltonian counterparts
- Applications include network design (computer networks), optimization problems (delivery routes), and DNA sequencing (genome assembly)
Conditions for Eulerian and Hamiltonian Graphs
Eulerian Path and Cycle Conditions
- has Eulerian cycle if all vertices have even degree
- Connected graph has Eulerian path if exactly zero or two vertices have odd degree
- Two odd-degree vertices serve as start and end points of path
- Directed graphs require for Eulerian cycle
- Every vertex must have equal in-degree and out-degree
- Directed graphs allow Eulerian path under specific degree conditions
- At most one vertex has (out-degree) - (in-degree) = 1
- At most one vertex has (in-degree) - (out-degree) = 1
- All other vertices have equal in-degree and out-degree
Hamiltonian Path and Cycle Conditions
- Necessary condition for Hamiltonian cycle requires connected graph with at least 3 vertices
- Sufficient conditions include Dirac's theorem and
- Dirac's theorem states minimum degree ≥ n/2 (n vertices)
- Ore's theorem requires sum of degrees of non-adjacent vertices ≥ n
- Graph closure concept introduced by Bondy and Chvátal aids in determining Hamiltonicity
- Involves adding edges between non-adjacent vertices with sufficient degree sum
Finding Eulerian and Hamiltonian Paths
Algorithms for Eulerian Paths and Cycles
- efficiently finds Eulerian cycles in undirected graphs
- Start at any vertex and follow unused edges until returning to start
- Recursively traverse any remaining circuits
- constructs Eulerian paths/cycles
- Choose non-bridge edges whenever possible
- Ensures all edges used exactly once
- extends Eulerian path finding
- Solved using matching algorithms for non-Eulerian graphs
Algorithms for Hamiltonian Paths and Cycles
- (DFS) finds Hamiltonian paths/cycles
- Worst-case time complexity exponential
- solves
- Uses dynamic programming
- Finds minimum-weight Hamiltonian cycle in O(n^2 * 2^n) time
- Approximation algorithms find near-optimal solutions in polynomial time
- for Traveling Salesman Problem
- Heuristic approaches quickly find good Hamiltonian cycles in large graphs
- commonly used in practice
Proofs for Eulerian and Hamiltonian Graphs
Eulerian Graph Proofs
- Prove connected graph has Eulerian cycle iff all vertices have even degree
- Use and construct cycle
- Demonstrate graph has Eulerian path iff exactly zero or two vertices have odd degree
- Consider degrees of start and end vertices
Hamiltonian Graph Proofs
- Prove Dirac's theorem for Hamiltonian cycles
- Show if G has n ≥ 3 vertices and minimum degree δ(G) ≥ n/2, then G Hamiltonian
- Establish Ore's theorem for Hamiltonian cycles
- Prove if G has n ≥ 3 vertices and d(u) + d(v) ≥ n for non-adjacent vertices u and v, then G Hamiltonian
- Prove on graph closure and Hamiltonicity
- Demonstrate determining Hamiltonian cycle NP-complete
- Reduce from 3-SAT problem
- Prove Tutte's theorem that every 4-connected planar graph Hamiltonian
- Use concept of Tutte cycles
Key Terms to Review (23)
Bondy-Chvátal Theorem: The Bondy-Chvátal Theorem is a fundamental result in graph theory that provides a necessary and sufficient condition for a graph to be Hamiltonian based on the lengths of cycles and paths. It specifically states that if a graph is connected and every pair of non-adjacent vertices has a common neighbor or the vertices are such that their distance within the graph satisfies certain criteria, then the graph must contain a Hamiltonian cycle. This theorem connects deeply to the concepts of Hamiltonian paths and cycles, emphasizing how they can be characterized through the relationships between vertices.
Chinese Postman Problem: The Chinese Postman Problem is a combinatorial optimization problem that seeks to find the shortest closed path or circuit that visits every edge of a graph at least once. This problem is particularly significant because it combines aspects of Eulerian paths and circuits, as well as real-world applications such as routing and network design.
Christofides Algorithm: The Christofides Algorithm is a heuristic method used to find approximate solutions to the Metric Traveling Salesman Problem (TSP). It guarantees a solution that is no more than 1.5 times the optimal solution length, making it a vital tool in combinatorial optimization. By combining concepts of minimum spanning trees and perfect matchings, it effectively finds Hamiltonian cycles, connecting the ideas of Eulerian and Hamiltonian paths and cycles.
Connected graph: A connected graph is a type of graph in which there is a path between every pair of vertices. This means you can travel from any vertex to any other vertex without having to leave the graph, which is essential for understanding many graph-related concepts like paths, cycles, and connectivity. Connected graphs help in analyzing various properties such as degree sequences and are crucial when exploring special types of paths and cycles, as well as determining the robustness of a graph's structure.
Degree of a Vertex: The degree of a vertex in a graph is the number of edges incident to it, reflecting how connected that vertex is within the graph. This concept helps understand various properties of graphs, including their structure, connectivity, and traversal capabilities. The degree is crucial for analyzing graph characteristics, such as the possibility of certain paths or cycles, and also plays a significant role in various graph coloring and planar properties.
Depth-first search: Depth-first search (DFS) is an algorithm used for traversing or searching through graphs and tree data structures by exploring as far along each branch as possible before backtracking. This method is particularly useful in exploring paths and cycles, determining Eulerian and Hamiltonian paths, assessing connectivity, and finding minimum spanning trees. It systematically visits vertices and edges, allowing for comprehensive exploration of graph structures.
Euler's Theorem: Euler's Theorem is a foundational principle in graph theory that states a connected graph can have an Eulerian circuit if and only if every vertex has an even degree. This concept is crucial for understanding the structure and properties of Eulerian paths and cycles, which are routes through a graph that visit every edge exactly once. In this context, it helps distinguish between graphs that can be traversed without lifting a pen versus those that cannot, influencing many applications in fields like network design and logistics.
Eulerian cycle: An Eulerian cycle is a closed path in a graph that visits every edge exactly once and returns to the starting vertex. This concept is pivotal in understanding how to traverse graphs and has practical applications in fields such as network design and route optimization. To have an Eulerian cycle, a connected graph must have all vertices of even degree, allowing for such a complete traversal without retracing any edges.
Eulerian Path: An Eulerian path is a trail in a graph that visits every edge exactly once. This concept plays a crucial role in understanding the structure of graphs and their connectivity, linking to various properties such as degrees of vertices and cycles. Eulerian paths can exist in graphs with specific configurations of vertex degrees, making them fundamental to problems in network design and route optimization.
Fleury's Algorithm: Fleury's Algorithm is a method used to find an Eulerian path or circuit in a graph by traversing each edge exactly once. The algorithm operates by ensuring that when possible, it traverses edges that do not lead to a dead end, thus maintaining the ability to continue the path. This algorithm highlights the practical application of Eulerian paths in graph theory, connecting concepts of connectivity and traversal in networks.
Hamiltonian Cycle: A Hamiltonian cycle is a path in a graph that visits each vertex exactly once and returns to the starting vertex, effectively creating a closed loop. Understanding Hamiltonian cycles is crucial in studying graph properties, particularly in relation to paths, cycles, and walks, where the emphasis is on how vertices and edges interact. The distinction between Hamiltonian and Eulerian paths becomes significant, as Hamiltonian cycles do not require all edges to be traversed, focusing solely on visiting each vertex uniquely.
Hamiltonian Path: A Hamiltonian path is a path in a graph that visits each vertex exactly once. It is named after the mathematician William Rowan Hamilton and is a crucial concept when analyzing the structure and characteristics of graphs, particularly in relation to paths, cycles, and walks. Understanding Hamiltonian paths helps to explore more complex structures like Hamiltonian cycles, which return to the starting vertex, and connects to broader principles of graph theory.
Handshaking Lemma: The handshaking lemma states that in any undirected graph, the sum of the degrees of all vertices is twice the number of edges. This result emphasizes the relationship between vertex degrees and edges, serving as a foundational concept in graph theory. It connects to various aspects such as degree sequences, path properties, and the structure of graphs.
Held-Karp Algorithm: The Held-Karp algorithm is a dynamic programming solution used to solve the Traveling Salesman Problem (TSP), which seeks the shortest possible route that visits each city exactly once and returns to the origin city. It efficiently computes the optimal solution by breaking the problem into smaller subproblems, using previously computed results to build towards the final answer. This algorithm is particularly significant in understanding Hamiltonian cycles, as it helps in determining the minimal cost Hamiltonian circuit in a complete graph.
Hierholzer's Algorithm: Hierholzer's Algorithm is a method used to find an Eulerian circuit or an Eulerian path in a graph. This algorithm is significant because it systematically constructs the Eulerian circuit by traversing edges and utilizing a stack to manage paths, ensuring that every edge is visited exactly once. Its efficiency and clear approach make it a fundamental tool when dealing with graphs that meet the necessary conditions for Eulerian trails.
Icosian Game: The Icosian Game is a mathematical puzzle that involves finding a Hamiltonian cycle on a graph formed by the vertices and edges of a dodecahedron, also known as an icosahedron. In this game, the player must determine a path that visits each vertex exactly once and returns to the starting point, showcasing principles related to Hamiltonian cycles and paths. This game not only serves as an engaging activity but also exemplifies significant concepts in graph theory and combinatorics.
Lin-kernighan heuristic: The lin-kernighan heuristic is an algorithm used for solving the traveling salesman problem (TSP) that focuses on improving the tour length through a series of local optimizations. This method builds on the basic 2-opt and 3-opt techniques but extends them to a more flexible approach that dynamically chooses which edges to remove and reintegrate in order to minimize the overall path. The effectiveness of this heuristic lies in its ability to find high-quality solutions quickly, making it valuable in both theoretical and practical applications.
Necessary conditions for Eulerian paths: Necessary conditions for Eulerian paths refer to specific criteria that must be satisfied for a graph to contain a path that visits every edge exactly once. These conditions relate to the degree of vertices in the graph, influencing the existence of such paths and thus connecting to the broader concepts of Eulerian and Hamiltonian paths and cycles.
Ore's Theorem: Ore's Theorem is a significant result in graph theory that provides a criterion for the existence of Hamiltonian cycles in a graph. Specifically, it states that if a graph has 'n' vertices (with n ≥ 3) and every pair of non-adjacent vertices has a combined degree of at least 'n', then the graph contains a Hamiltonian cycle. This theorem connects with the properties of both Eulerian and Hamiltonian paths and cycles by addressing conditions under which these cycles can be found.
Seven Bridges of Königsberg: The Seven Bridges of Königsberg is a famous problem in mathematics and graph theory that asks whether it is possible to traverse all seven bridges connecting the islands of the city of Königsberg without crossing any bridge more than once. This problem led to the development of important concepts like Eulerian paths, where a path visits every edge exactly once, and sparked further investigations into the nature of networks and connectivity.
Strongly connected digraph: A strongly connected digraph is a directed graph where there is a directed path from every vertex to every other vertex. This property is crucial when analyzing the connectivity of directed graphs, particularly when considering paths and cycles that visit multiple vertices. In the context of Eulerian and Hamiltonian paths, the concept of strong connectivity ensures that there are potential routes that allow traversal through the entire graph, making it essential for understanding the existence of certain types of paths and cycles.
Sufficient conditions for hamiltonian cycles: Sufficient conditions for Hamiltonian cycles are specific criteria that, when met, guarantee the existence of a Hamiltonian cycle in a graph. These conditions help in identifying whether a graph contains a cycle that visits each vertex exactly once before returning to the starting vertex, a crucial concept in combinatorial optimization and graph theory.
Traveling Salesman Problem: The Traveling Salesman Problem (TSP) is a classic optimization problem that asks for the shortest possible route that visits a set of cities exactly once and returns to the origin city. This problem is a cornerstone in combinatorial optimization and is closely related to Hamiltonian cycles, which involve visiting each vertex exactly once in a graph, and it showcases challenges in finding efficient solutions within graphs.