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A Hamiltonian cycle around a network of six vertices Examples of Hamiltonian cycles on a square grid graph 8x8. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.
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 ...
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.
A path such that no graph edges connect two nonconsecutive path vertices is called an induced path. A path that includes every vertex of the graph without repeats is known as a Hamiltonian path. Two paths are vertex-independent (alternatively, internally disjoint or internally vertex-disjoint) if they do not have any internal vertex or edge in ...
The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. [3] Rédei's theorem is the special case for complete graphs of the Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to the chromatic number of these graphs. [4]
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 ...
The paths can then all be completed to Hamiltonian cycles by connecting their ends through the remaining vertex. [ 2 ] Expanding a vertex of a 2 k {\displaystyle 2k} -regular graph into a clique of 2 k {\displaystyle 2k} vertices, one for each endpoint of an edge at the replaced vertex, cannot change whether the graph has a Hamiltonian ...
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