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  2. Hamiltonian path problem - Wikipedia

    en.wikipedia.org/wiki/Hamiltonian_path_problem

    A verifier algorithm for Hamiltonian path will take as input a graph G, starting vertex s, and ending vertex t. Additionally, verifiers require a potential solution known as a certificate, c. For the Hamiltonian Path problem, c would consist of a string of vertices where the first vertex is the start of the proposed path and the last is the end ...

  3. Hamiltonian path - Wikipedia

    en.wikipedia.org/wiki/Hamiltonian_path

    A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

  4. Path (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Path_(graph_theory)

    A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).

  5. Lovász conjecture - Wikipedia

    en.wikipedia.org/wiki/Lovász_conjecture

    Another version of Lovász conjecture states that . Every finite connected vertex-transitive graph contains a Hamiltonian cycle except the five known counterexamples.. There are 5 known examples of vertex-transitive graphs with no Hamiltonian cycles (but with Hamiltonian paths): the complete graph, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter ...

  6. Eulerian path - Wikipedia

    en.wikipedia.org/wiki/Eulerian_path

    Following the edges in alphabetical order gives an Eulerian circuit/cycle. In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex.

  7. Ore's theorem - Wikipedia

    en.wikipedia.org/wiki/Ore's_theorem

    Illustration for the proof of Ore's theorem. In a graph with the Hamiltonian path v 1...v n but no Hamiltonian cycle, at most one of the two edges v 1 v i and v i − 1 v n (shown as blue dashed curves) can exist. For, if they both exist, then adding them to the path and removing the (red) edge v i − 1 v i would produce a Hamiltonian cycle.

  8. Tournament (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Tournament_(graph_theory)

    a is inserted between v 2 and v 3.. Any tournament on a finite number of vertices contains a Hamiltonian path, i.e., directed path on all vertices (Rédei 1934).. This is easily shown by induction on : suppose that the statement holds for , and consider any tournament on + vertices.

  9. Knight's tour - Wikipedia

    en.wikipedia.org/wiki/Knight's_tour

    The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time. [4]