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#Euler

#Types of Graph: #Euler and #Hamiltonian Graphs:~ In graph theory, different types of graphs are defined based on how vertices and edges are connected and traversed. Some important types related to traversal are Euler Walk, Euler Trail, Euler Path, Euler Circuit, Euler Graph, and Hamiltonian Graph. #An Euler Walk is a walk in a graph where edges may be repeated, but the walk follows connected edges from one vertex to another. #An Euler Trail is a trail that uses every edge of the graph exactly once, but the starting and ending vertices may be different. #An Euler Path is similar to an Euler Trail; it is a path that passes through each edge exactly once. A graph has an Euler Path if it has exactly two vertices of odd degree. #An Euler Circuit is a closed trail that starts and ends at the same vertex and uses every edge exactly once. A graph has an Euler Circuit if all vertices have even degree. #An Euler Graph is a connected graph that contains an Euler Circuit. In such graphs, every vertex has an even degree. #A Hamiltonian Graph is a graph that contains a Hamiltonian Path or Hamiltonian Circuit. A Hamiltonian Path visits each vertex exactly once, while a Hamiltonian Circuit starts and ends at the same vertex after visiting every vertex exactly once. #In summary, Euler concepts focus on edges, while Hamiltonian concepts focus on vertices, making both essential ideas in graph theory and discrete mathematics.