Difference Between Eulerian And Hamiltonian Graph

Traversing the Labyrinth: Eulerian vs. Hamiltonian Graphs

Imagine you're standing before a sprawling, complex network – a city map, a computer chip, or even the intricate web of connections in the human brain. Each intersection or component is a node, and each road or link is an edge. Navigating these networks efficiently is crucial, and understanding the fundamental differences between Eulerian and Hamiltonian graphs unlocks the secrets to doing so. These two graph types represent distinct yet fascinating approaches to traversing networks, each with its own unique properties and applications. This article will illuminate the distinctions between these important concepts, unraveling the mysteries of Eulerian and Hamiltonian paths and circuits.

What is a Graph?

Before diving into Eulerian and Hamiltonian graphs, let's establish a basic understanding of what a graph is. In graph theory, a graph is a visual representation of connections between objects. These objects are represented as nodes (also called vertices), and the connections between them are represented as edges. Think of a road map: cities are nodes, and roads are edges. Graphs can be directed (edges have a specific direction) or undirected (edges can be traversed in either direction).

Eulerian Graphs: The Path of the Bridges

A graph is considered Eulerian if it contains an Eulerian circuit or an Eulerian path.

Eulerian Circuit: An Eulerian circuit is a closed path (a path that starts and ends at the same node) that traverses every edge of the graph exactly once. Think of trying to walk across every bridge in a city exactly once, returning to your starting point.

Eulerian Path: An Eulerian path is an open path (a path that starts and ends at different nodes) that traverses every edge exactly once. This is like walking across every bridge in a city exactly once, but not returning to your starting point.

The Key Condition: A connected graph (meaning you can get from any node to any other node) possesses an Eulerian circuit if and only if every node has an even degree (an even number of edges connected to it). If exactly two nodes have an odd degree, an Eulerian path exists. If more than two nodes have an odd degree, neither an Eulerian circuit nor an Eulerian path is possible.

Real-world Application: The famous Königsberg Bridge problem, solved by Leonhard Euler, is a classic example. The problem involved determining whether it was possible to traverse all seven bridges of Königsberg exactly once and return to the starting point. Euler proved it was impossible because several nodes (landmasses) had odd degrees. Eulerian graphs are also used in network optimization, designing efficient routes for garbage collection, and planning street sweeping routes.

Hamiltonian Graphs: The Quest for Complete Coverage

A graph is considered Hamiltonian if it contains a Hamiltonian cycle or a Hamiltonian path.

Hamiltonian Cycle: A Hamiltonian cycle is a closed path that visits every node of the graph exactly once. Imagine trying to visit every major city in a country, returning to your starting city, without visiting any city twice.

Hamiltonian Path: A Hamiltonian path is an open path that visits every node of the graph exactly once. This is similar to the Hamiltonian cycle, but you don't need to return to the starting point.

The Key Challenge: Unlike Eulerian graphs, there's no simple rule to determine if a graph is Hamiltonian. Finding a Hamiltonian cycle or path is an NP-complete problem, meaning there's no known algorithm that can solve it efficiently for large graphs. This makes finding Hamiltonian paths computationally challenging.

Real-world Applications: Hamiltonian graphs find applications in various fields, including DNA sequencing, robotics (planning optimal robot paths), and the Traveling Salesperson Problem (TSP), a classic optimization problem where a salesperson needs to find the shortest route that visits all cities and returns to the origin.

Key Differences Summarized:

| Feature | Eulerian Graph | Hamiltonian Graph |
|-----------------|-------------------------------------------------|-------------------------------------------------|
| Focus | Edges | Nodes |
| Condition | Even degree nodes (for circuits) | No simple, easily verifiable condition |
| Path/Circuit | Traverses every edge exactly once | Visits every node exactly once |
| Computational Complexity | Relatively easy to determine | NP-complete, computationally hard |
| Real-world examples | Network optimization, Königsberg bridges | Traveling Salesperson Problem, DNA sequencing |

Conclusion:

Eulerian and Hamiltonian graphs represent distinct but equally important approaches to navigating networks. While Eulerian graphs focus on traversing every edge, Hamiltonian graphs emphasize visiting every node. Understanding these distinctions opens doors to efficient solutions in various fields, ranging from logistics and route planning to computational biology and network design. The contrast between the easily verifiable conditions for Eulerian graphs and the computationally challenging nature of determining Hamiltonian graphs highlights the rich and complex landscape of graph theory.

FAQs:

1. Can a graph be both Eulerian and Hamiltonian? Yes, a graph can be both Eulerian and Hamiltonian. However, this is not always the case.

2. What if a graph is disconnected? The definitions of Eulerian and Hamiltonian paths/circuits only apply to connected graphs.

3. Are there algorithms to find Hamiltonian paths/cycles? While there isn't a fast algorithm guaranteed to find a Hamiltonian path/cycle in all cases, several heuristics and approximation algorithms exist for finding likely solutions or good approximations.

4. What is the significance of NP-completeness in the context of Hamiltonian graphs? NP-completeness means that the problem of finding a Hamiltonian cycle is computationally hard, meaning the time it takes to solve the problem grows exponentially with the size of the graph. This makes finding solutions for large graphs incredibly difficult.

5. Beyond TSP, what other practical applications exist for Hamiltonian graphs? Applications extend to tasks such as scheduling, robotics path planning, designing efficient communication networks, and even tasks in logistics and supply chain management where complete coverage of locations is necessary.

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Chapter 7.2 Euler Path and Hamiltonian Circuit - University of … An Euler path in a graph G is a path that uses each arc of G exactly once. We want to find if an Euler path exists in graph G. Whether an Euler path exists in a given graph hinges on the degrees of its nodes. A node is even if its degree is even and odd if its degree is odd. THEOREM ON ODD NODES IN A GRAPH

Hamiltonian and Eulerian Graphs - University of South Carolina A spanning cycle in a graph is called a Hamiltonian cycle, and a spanning path is called a Hamiltonian path. A graph is said to be Hamiltonian if it has a spanning cycle and it is said to be traceable if it has a Hamiltonian path. The graph above, known as the dodecahedron, was the basis for a game

Lecture 12 - Eulerian trails and circuits. Hamiltonian paths and A connected graph G contains an Eulerian trail if and only if there are at most two vertices of odd degree. Isabela Dr amnesc UVT Graph Theory and Combinatorics { Lecture 12 10/24

Eulerian and Hamiltonian Cycles - University of Pennsylvania De ̄nition 1.0.1 Given a graph G, an Eule-rian cycle is a cycle in G that passes through all the nodes (possibly more than once) and every edge of G exactly once. A Hamiltonian cycle is a cycle that passes through all the nodes exactly once (note, some edges may not be traversed at all).

ON EULERIAN AND HAMILTONIAN GRAPHS AND LINE GRAPHS An eulerian walk of G is a spanning tour containing every line of G. A cycle of G is a closed walk v « x, ,v_,X-, . .. , v ,x ,v in which v.,..., v are 1122 nnl 1 n distinct and n >• 3. An hamiltonian cycle of G contains every point of G. A graph is eulerian if it has an eulerian walk; it is hamiltonian if it has an hamiltonian cycle.

Comprehensive Study of Eulerian and Hamiltonian Graphs Among the various concepts in graph theory, Eulerian and Hamiltonian graphs hold significant importance due to their theoretical richness and practical applicability. An Eulerian graph is one in which there exists a trail that traverses each edge exactly once, …

Eulerian and Hamiltonian Graphs. - cs.rpi.edu 1 Eulerian and Hamiltonian Graphs. Definition. A connected graph is called Eulerian if it has a closed trail containing all edges of the graph. D .. Question 1: What is the necessary and sufficient con-dition for a graph to be Eulerian? Question 2: Is there a fast algorithm to construct an Eulerian trail if it exists? V (G).

Lecture 24, Euler and Hamilton Paths - Duke University A simple path in a graph G that passes through every vertex exactly once is called a Hamilton path, and a simple circuit in a graph G that passes through every vertex exactly once is called a Hamilton circuit.

9.4 Traversal: Eulerian and Hamiltonian Graphs Definitions: Eulerian Paths, Circuits, Graphs. A Eulerian path through a graph is a path whose edge list contains each edge of the graph exactly once. If the path is a circuit, then it is called a Eulerian circuit. A Eulerian graph is a graph that possesses a Eulerian path. Example 9.4.1.

Eulerian and Hamiltonian Paths - uoc.gr Definition 1: An Euler path is a path that crosses each edge of the graph exactly once. If the path is closed, we have an Euler circuit. In order to proceed to Euler’s theorem for checking the existence of Euler paths, we define the notion of a vertex’s degree.

UNIT 3 EULERIAN AND HAMILTONIAN GRAPHS - eGyanKosh E1) Prove that the graph given in Fig.5(c) is Eulerian by producing an Eulerian circuit in it. E2) What is the difference between an Eulerian graph and an Eulerian circuit?

What isthe di erence between Eulerian and Hamiltonian graphs? I A graph is Eulerian )Every vertex has even degree Easy to check I There are very e ective algorithms to construct Eulerian cycles < O(jEj2) I To determine whether a graph is Hamiltonian is NP-complete Very hard I The algorithms to construct Hamiltonian cycles …

2. EULERIAN AND HAMILTONIAN GRAPHS - coopersnotes.net An Eulerian cycle in a graph (undirected with no multiple edges) is one that passes along every edge exactly once. An Eulerian graph is one that has an Eulerian cycle. (Remember to pronounce Eulerian as “Oil-air-ian”.) Example 2: Which of the following graphs are Eulerian? Solution: A is Eulerian, but not B or C. The third graph

Circuits - (Eulerian and Hamiltonian) Let be a connected graph. 1 has an Eulerian circuit (i.e., is Eulerian) if and only if every vertex of has even degree. 2 has an Eulerian path, but not an Eulerian circuit, if and only if has exactly two vertices of odd degree. I The Eulerian path in this case must start at any of the two ’odd-degree’

3 Euler Circuits and Hamilton Cycles - UCD An Euler circuit in a graph is a circuit which includes each edge exactly once. An Euler trail is a walk which contains each edge exactly once, i.e., a trail which includes every edge. A Hamilton cycle is a cycle in a graph which contains each vertex exactly once. A connected graph is called Hamiltonian is it contains a Hamilton cycle. There

Eulerian graphs - cvut.cz Such a cycle is a Hamiltonian cycle and G is a Hamiltonian graph. A non-Hamiltonian graph G is semi-Hamiltonian if there exists a path passing through every vertex (not closed). Figs 2.1, 2.2 and 2.3 show graphs that are Hamiltonian, semi-Hamiltonian and non-Hamiltonian, respectively.

Eulerian and Hamiltonian Cycles - Polytechnique Montréal A cycle that travels exactly once over each edge in a graph is called “Eulerian.” A cycle that travels exactly once over each vertex in a graph is called “Hamiltonian.” Some graphs possess neither a Hamiltonian nor a Eulerian cycle, such as the one below.

3. Eulerian and HamiltonianGraphs - ELTE also resulted in the special types of graphs, now called Eulerian graphs and Hamiltonian graphs. Due to the rich structure of these graphs, they ﬁnd wide use both in research and application. 3.1 Euler Graphs A closed walk in a graph G containing all the edges of G is called an Euler line in G. A graph containingan Euler line is called an ...

Eulerian and Hamiltonian Graphs - Stanford University Theorem: If G is a connected graph where every node has even degree, then G is Eulerian. Proof: Consider the longest path P in G with no repeated edges. We claim that P is an Eulerian cycle. To see why, suppose not. If P isn't a cycle, then by Lemma 1, we can extend P into a longer cycle P', contradicting that P is the longest path in the graph.

Euler Paths, Planar Graphs and Hamiltonian Paths An undirected graph has an Eulerian path if and only if exactly zero or two vertices have odd degree

Difference Between Eulerian And Hamiltonian Graph

Traversing the Labyrinth: Eulerian vs. Hamiltonian Graphs

What is a Graph?

Eulerian Graphs: The Path of the Bridges

Hamiltonian Graphs: The Quest for Complete Coverage

Key Differences Summarized:

Conclusion:

FAQs:

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