Euler trails are fascinating paths in graphs that use every edge exactly once. They're like drawing a shape without lifting your pen or planning a city tour that covers every street without repeating.
Bridges and local bridges play a key role in Euler trails. Understanding vertex degrees helps determine if a graph has an Euler trail or circuit. Fleury's algorithm is a practical way to find these paths in graphs.
Euler Trails
Bridges and local bridges
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Bridges are edges in a graph that disconnect the graph into two or more components when removed (London Bridge , Golden Gate Bridge )
Removing a bridge increases the number of connected components
Graphs with bridges cannot have Euler circuits because removing the bridge breaks the path
Local bridges are edges that create new bridges when removed but do not disconnect the graph (Tower Bridge , Brooklyn Bridge )
Removing a local bridge creates a new bridge in the graph
Graphs with local bridges can have Euler trails but not Euler circuits since the local bridge must be traversed last
Application of Fleury's algorithm
Fleury's algorithm finds Euler trails or circuits in graphs by following these steps:
Choose a starting vertex (if two vertices have odd degree , start at one of them)
Follow an edge to an adjacent vertex, erasing the edge as you go (like drawing a path without lifting the pen)
Stop if no edges are left
Choose a non-bridge edge if possible, otherwise use the only remaining edge
Repeat steps 2-4 until all edges are traversed (like completing a maze)
The resulting path is an Euler trail or circuit depending on the starting and ending vertices (Königsberg bridge problem , Seven Bridges of Königsberg )
Euler trails and vertex degrees
Vertex degree is the number of edges connected to a vertex (like the number of roads leading into a city)
Euler trails are paths that use every edge in a graph exactly once (like walking every street in a city without repeating)
Graphs have Euler trails if they are connected and have either zero or two vertices with odd degree
If two vertices have odd degree, they must be the start and end points of the Euler trail (like starting and ending a city tour at specific locations)
Euler circuits are Euler trails that start and end at the same vertex (like a circular city tour)
Graphs have Euler circuits if they are connected and all vertices have even degree (like every city having an even number of roads leading in and out)
To determine if a graph has an Euler trail or circuit:
Check if the graph is connected (there is a path between every pair of vertices)
Count the vertices with odd degree
No odd degree vertices → \rightarrow → Euler circuit exists
Exactly two odd degree vertices → \rightarrow → Euler trail exists
More than two odd degree vertices → \rightarrow → No Euler trail or circuit exists
Graph theory is the mathematical study of structures consisting of vertices and edges, which provides the foundation for understanding Euler trails
Traversability refers to the possibility of traversing all edges in a graph, which is a key concept in Euler trails
A path is a sequence of vertices connected by edges, while a cycle is a path that starts and ends at the same vertex
Connectivity in a graph ensures that there is a path between any two vertices, which is crucial for the existence of Euler trails