Difference between hamiltonian path and euler path

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Can some one tell me the difference between hamiltonian path and euler path  They seem similar

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An Euler path is a path that passes through every edge exactly once  If it ends at the initial vertex then it is an Euler cycle    A Hamiltonian path is a path that passes through every vertex exactly once  NOT every edge   If it ends at the initial vertex then it is a Hamiltonian cycle    In an Euler path you might pass through a vertex more than once    In a Hamiltonian path you may not pass through all edges

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An Euler path is a path that uses every edge of a  graph exactly once and it must have exactly two odd vertices the path starts and ends at different vertex                                                           A Hamiltonian cycle is a cycle that contains every vertex of the graph hence you may not use all the edges of the graph

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Eulerian path must visit each edge exactly once  while Hamiltonian path must visit each vertex exactly once

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Euler path is a graph using every edge NOTE  of the graph exactly once  Euler circuit is a euler path that returns to it starting point after covering all edges   While hamilton path is a graph that covers all vertex NOTE  exactly once  When this path returns to its starting point than this path is called hamilton circuit

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Euler Path - An Euler path is a path in which each edge is traversed exactly once   Hamiltonian Path - An Hamiltonian path is path in which each vertex is traversed exactly once   If you have ever confusion remember E - Euler E - Edge

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They are related but are neither dependent nor mutually exclusive  If a graph has an Eurler cycle  it may or may not also have a Hamiltonian cyle and vice versa     Euler cycles visit every edge in the graph exactly once  If there are vertices in the graph with more than two edges  then by definition  the cycle will pass through those vertices more than once  As a result  vertices can be repeated but edges cannot   Hamiltonian cycles visit every vertex in the graph exactly once  similar to the travelling salesman problem    As a result  neither edges nor vertices can be repeated

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I ll use a common example in biology  reconstructing a genome by making DNA samples   De-novo assembly  To construct a genome from short reads  it s necessary to construct a graph of those reads  We do it by breaking the reads into k-mers and assemble them into a graph     We can reconstruct the genome by visiting each node once as in the diagram  This is known as Hamiltonian path   Unfortunately  constructing such path is NP-hard  It s not possible to derive an efficient algorithm for solving it  Instead  in bioinformatics we construct a Eulerian cycle where an edge represents an overlap

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A Hamiltonian path visits every node  or vertex  exactly once  and a Eulerian path traverses every edge exactly once

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Graph Theory Definitions   In descending order of generality    Walk  a sequence of edges where the end of one edge marks the beginning of the next edge Trail  a walk which does not repeat any edges  All trails are walks  Path  a walk where each vertex is traversed at most once   paths used to refer to open walks  the definition has changed now  The property of traversing vertices at most once means that edges are also crossed at most once  hence all paths are trails    Hamiltonian paths  amp  Eulerian trails   Hamiltonian path  visits every vertex in the graph  exactly once  because it is a path  Eulerian trail  visits every edge in the graph exactly once  because it is a trail  vertices may well be crossed more than once

[algorithm] Difference between hamiltonian path and euler path

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