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

• 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:
  1. Choose a starting vertex (if two vertices have odd degree, start at one of them)
  2. Follow an edge to an adjacent vertex, erasing the edge as you go (like drawing a path without lifting the pen)
  3. Stop if no edges are left
  4. Choose a non-bridge edge if possible, otherwise use the only remaining edge
  5. 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:
  1. Check if the graph is connected (there is a path between every pair of vertices)
  2. 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 Concepts Related to Euler Trails

• 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