12.5 Euler Circuits
3 min read•june 18, 2024
Graphs are powerful tools for modeling connections. They can represent anything from social networks to transportation systems. Understanding helps us analyze how different parts of a system relate to each other.
Euler circuits are special paths that use every in a graph exactly once. They have practical applications in route planning and network design. Knowing how to identify and construct Euler circuits can help solve real-world problems efficiently.
Graph Connectivity
Connectivity of graphs
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- A graph is connected if there is a between every pair of vertices
- Path represents a sequence of edges that begins at a of a graph and travels from vertex to vertex along edges of the graph (highway network connecting cities)
- A graph that is not connected is disconnected
- There exists at least one pair of vertices with no path between them (remote island with no bridge or ferry to mainland)
- A consists of a single
- Component refers to a connected subgraph of a graph (contiguous land mass)
- A has two or more components
- Vertices of one component have no paths to vertices of another component (separate social cliques with no mutual friends)
Euler Circuits
Euler circuit theorem application
- An is a that uses every of a graph exactly once
- denotes a path that begins and ends at the same vertex (round-trip flight itinerary)
- A graph is called Eulerian if it contains an Euler circuit
- states that a connected graph is Eulerian if and only if each vertex has
- of a vertex represents the number of edges incident with it (number of roads leading into an intersection)
- An even degree vertex has a degree that is a multiple of 2 (4-way stop sign)
- If a graph has one or more vertices with , it is not Eulerian
- An odd degree vertex has a degree that is not a multiple of 2 (dead-end street)
- The concept of Euler circuits was first explored in the
Construction of Euler circuits
- To find an Euler circuit in an :
- Choose any vertex to start, and trace a circuit back to this vertex, using as many unused edges as possible
- If any unused edges remain, start a new circuit at a vertex in the previous circuit that has unused edges
- Trace this new circuit back to its starting point, using only previously unused edges
- Combine the circuits by starting at the vertex where the second circuit started, follow the second circuit, and then follow the first circuit
- Repeat steps 2-4 if unused edges remain, starting at any vertex in the combined circuit that has unused edges
- Always trace the new circuit first when combining (draw inner loop before connecting outer loop)
- The process ends when no unused edges remain, resulting in an Euler circuit (single continuous line that covers entire graph)
Advanced Graph Concepts
Graph representations and properties
- : A square matrix used to represent a finite graph, where the elements indicate whether pairs of vertices are adjacent or not
- : A graph that can be drawn on a plane without any edges crossing
- : A bijective correspondence between the vertices of two graphs that preserves the edge connectivity
- : A path in an undirected or directed graph that visits each vertex exactly once and returns to the starting vertex
Key Terms to Review (22)
Adjacency matrix: An adjacency matrix is a square grid used to represent a finite graph, where each cell indicates whether pairs of vertices are adjacent or not in the graph. This matrix provides a way to store and analyze graph structures efficiently, making it easier to compare different graphs and investigate properties like connectivity and pathfinding. In particular, it plays a crucial role in algorithms used to determine Euler circuits, as it simplifies the identification of edges connecting vertices.
Circuit: A circuit in graph theory is a path that starts and ends at the same vertex with no other repeated vertices. It is a closed loop in a graph where each edge is used exactly once.
Circuit: A circuit is a closed path through which an electric current flows or a sequence of connected edges in a graph that starts and ends at the same vertex. In the context of navigating graphs, circuits are important for determining routes and connections, while in Euler circuits, they represent paths that visit every edge exactly once before returning to the starting point. Understanding circuits is key to solving problems related to traversal and connectivity within graphs.
Component: In the context of Euler Circuits, a component refers to a connected subgraph within a larger graph where there is a path between every pair of vertices. Understanding components is crucial for determining whether an Euler Circuit exists, as it involves examining the connectivity and degree of each vertex in relation to the entire graph. In essence, the concept of a component helps identify distinct sections of a graph that can affect the existence of Euler Circuits.
Connected graph: A connected graph is a type of graph where there is a path between every pair of vertices, meaning all vertices are reachable from one another. This property is essential in understanding various concepts in graph theory, particularly in analyzing routes, networks, and the structure of trees. Connected graphs can have implications for finding Euler circuits and trails, as well as play a critical role in understanding the structure and properties of trees.
Degree: In mathematics, a degree is a unit of measurement for angles, denoting the size of the angle in a circle. It also represents the number of edges connected to a vertex in graph theory, providing insight into the structure and navigation of graphs. Understanding degrees helps clarify relationships between angles and graph structures, which is essential for analyzing various mathematical concepts.
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.
Disconnected graph: A disconnected graph is a type of graph where at least one pair of vertices does not have a path connecting them. This means that the graph can be split into two or more separate components, each of which is connected internally but not to each other. Understanding this concept is crucial, especially when discussing Euler circuits, as Euler circuits require the entire graph to be connected in order to traverse every edge exactly once without lifting the pencil.
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 circuit: An Euler circuit is a trail in a graph that visits every edge exactly once and returns to the starting vertex. This concept is tied to specific properties of a graph, particularly the degrees of its vertices. For a graph to have an Euler circuit, all vertices must have even degrees, which allows for a continuous path that does not lift the pen off the page.
Euler's Circuit Theorem: Euler's Circuit Theorem states that a connected graph can have an Euler circuit if and only if every vertex in the graph has an even degree. This theorem is significant in understanding the conditions under which a path can be traced through a graph such that every edge is traversed exactly once and the path ends at the starting vertex.
Eulerian graph: An Eulerian graph is a type of graph that contains an Euler circuit, meaning it has a closed trail that visits every edge exactly once. These graphs are significant because they help us understand connectivity and traversal within various structures, such as networks and paths. The existence of an Eulerian graph is determined by the degrees of its vertices, which can lead to applications in real-world problems like route planning and network design.
Even Degree: An even degree in graph theory refers to a vertex that has an even number of edges connected to it. This characteristic plays a significant role in determining the existence of Euler circuits and helps to identify Hamilton paths. Understanding even degree is crucial for recognizing patterns in graph connectivity and analyzing the structure of graphs.
Graph connectivity: Graph connectivity refers to the minimum number of elements (like vertices or edges) that need to be removed to disconnect the remaining vertices from one another. This concept is crucial because it helps to determine how resilient a graph is to disruptions and how information flows through it. The degree of connectivity can reveal a lot about the overall structure of the graph, influencing aspects like the existence of Euler circuits and making it essential when comparing different graphs.
Graph Isomorphism: Graph isomorphism is a concept in graph theory where two graphs can be considered the same if there exists a one-to-one correspondence between their vertices and edges, preserving the connectivity of the graphs. This means that two graphs are isomorphic if you can relabel the vertices of one graph to match the structure of the other graph without changing its connectivity. Understanding graph isomorphism is essential for studying various properties of graphs, including those related to Euler circuits, as it helps in identifying when different representations yield the same underlying structure.
Hamiltonian circuit: A Hamiltonian circuit is a path in a graph that visits each vertex exactly once and returns to the starting vertex, effectively forming a closed loop. This concept is crucial in various applications, such as routing and optimization problems, where visiting every point without repetition is necessary. Understanding Hamiltonian circuits helps in exploring graph theory's broader implications, particularly when contrasting with Euler circuits, which differ in the conditions for traversing edges rather than vertices.
Königsberg bridge problem: The Königsberg bridge problem is a famous mathematical puzzle that originated in the 18th century, involving the task of finding a walk through the city of Königsberg that would cross each of its seven bridges exactly once. This problem led to significant developments in graph theory and is foundational in understanding Euler Circuits and Trails.
Odd degree: In graph theory, an odd degree refers to a vertex that has an odd number of edges connected to it. This concept is significant in understanding the properties of graphs, particularly when analyzing Euler circuits, which involve traversing every edge exactly once. The presence of vertices with odd degrees influences the existence and structure of these circuits, as specific conditions must be met for a graph to contain an Euler circuit.
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.
Planar graph: A planar graph is a type of graph that can be drawn on a plane without any edges crossing each other. This means that the vertices and edges can be arranged in such a way that no two edges intersect except at their endpoints. Planar graphs have unique properties that make them important in understanding the structure of networks and in solving problems related to connectivity and routing.
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.