Search results
Results from the WOW.Com Content Network
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.
The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. They remain NP-complete even for special kinds of graphs, such as: bipartite graphs, [12]
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.
The term cycle may also refer to an element of the cycle space of a graph. There are many cycle spaces, one for each coefficient field or ring. The most common is the binary cycle space (usually called simply the cycle space), which consists of the edge sets that have even degree at every vertex; it forms a vector space over the two-element field.
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 ...
An early example of answer set programming was the planning method ... The following Lparse program can be used to find a Hamiltonian cycle in a given directed graph ...
There is always a Hamiltonian cycle in the wheel graph and there are + cycles in W n (sequence A002061 in the OEIS). The 7 cycles of the wheel graph W 4 . For odd values of n , W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color.
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.