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An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once
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 ...
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 significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
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 circuit may refer to a closed trail or an element of the cycle space (an Eulerian spanning subgraph). The circuit rank of a graph is the dimension of its cycle space. circumference The circumference of a graph is the length of its longest simple cycle. The graph is Hamiltonian if and only if its circumference equals its order. class 1.
A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2] Its eigenvalue will be the constant degree of the 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.