How do you prove a Hamiltonian circuit?

If every vertex of G has degree ≥ |V (G)|/2, then G has a Hamiltonian cycle. Proof: Assume that G satisisfies the condition, but does not have a Hamiltonian cycle. If it is possible to add edges to G so that the result still a simple graph with no Hamiltonian cycle, do so.

Does a cube have a Hamiltonian cycle?

Another example which always yields a hamiltonian graph is the cube function. In fact, if x is any line in a connected graph G with at least three points, then the cube of G has a hamiltonian cycle containing x.

How do you know if a Hamilton circuit exists?

Hamilton Path A Hamilton path is a simple path that traverses every vertex in G exactly once. If G is a simple graph with n vertices with n ≥ 3 such that deg(u) + deg(v) ≥ n for every pair of nonadjacent vertices u and v in G, then G has a Hamilton circuit.

How do you prove there is no Hamiltonian circuit?

Proving a graph has no Hamiltonian cycle [closed]

1. A graph with a vertex of degree one cannot have a Hamilton circuit.
2. Moreover, if a vertex in the graph has degree two, then both edges that are incident with this vertex must be part of any Hamilton circuit.
3. A Hamilton circuit cannot contain a smaller circuit within it.

For which N does KN contain a Hamilton path a Hamilton cycle explain?

For all n ≥ 3, Kn will contain a Hamilton cycle.

What do you mean by Hamiltonian cycle?

A Hamiltonian cycle is a closed loop on a graph where every node (vertex) is visited exactly once. A loop is just an edge that joins a node to itself; so a Hamiltonian cycle is a path traveling from a point back to itself, visiting every node en route.

What is Qn in graph theory?

In graph theory, the hypercube graph Qn is the graph formed from the vertices and edges of an n-dimensional hypercube. Qn has 2n vertices, 2n−1n edges, and is a regular graph with n edges touching each vertex.

How do you find the weight of a Hamilton circuit?

The total weight of a Hamilton circuit is the sum of the weights of all the edges in that circuit.

Which of the following graph has a Hamilton circuit?

Any connected graph that contains a Hamiltonian circuit is called as a Hamiltonian Graph.

For which values of m and n does km N have a Hamilton circuit justify your answer?

A circuit necessarily has 2k vertices for some positive integer k; k of these vertices are in V0, and the other k are in V1. Thus, if m≠n it is impossible for a circuit in Km,n to hit every vertex, and therefore Km,n can have a Hamilton circuit only if m=n.