Search results
Results From The WOW.Com Content Network
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 . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be ...
The requirement of returning to the starting city does not change the computational complexity of the problem; see Hamiltonian path problem. Another related problem is the bottleneck travelling salesman problem: Find a Hamiltonian cycle in a weighted graph with the minimal weight of the weightiest edge.
Because the Hamiltonian path problem is NP-complete, this reduction shows that the decision version of the longest path problem is also NP-complete. In this decision problem, the input is a graph G and a number k ; the desired output is yes if G contains a path of k or more edges, and no otherwise.
Alternatively, the problem can be solved by performing a binary search or sequential search for the smallest x such that the subgraph of edges of weight at most x has a Hamiltonian cycle. This method leads to solutions whose running time is only a logarithmic factor larger than the time to find a Hamiltonian cycle. [1]
The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. [5] 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. [6]
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.
A Hamiltonian cycle on a tesseract with vertices labelled with a 4-bit cyclic Gray code. Every hypercube Q n with n > 1 has a Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path exists between two vertices u and v if and only if they have different colors in a 2-coloring of the graph.