Is every Cayley graph Hamiltonian?

Every finite connected Cayley graph contains a Hamiltonian cycle. the conjecture holds rather than attack it in full generality.

How do you know if a graph is Hamiltonian?

A simple graph with n vertices in which the sum of the degrees of any two non-adjacent vertices is greater than or equal to n has a Hamiltonian cycle.

Are all regular graphs Hamiltonian?

Every connected k-regular graph on at most 2k + 2 vertices is Hamiltonian. Furthermore, we characterize connected k-regular graphs on 2k +3 vertices (when k is even) and 2k + 4 vertices (when k is odd) that are non-Hamiltonian.

What does it mean if a graph is Hamiltonian?

Definition: A graph is considered Hamiltonian if and only if the graph has a cycle containing all of the vertices of the graph. Definition: A Hamiltonian cycle is a cycle that contains all vertices in a graph . If a graph has a Hamiltonian cycle, then the graph is said to be Hamiltonian.

How do you make a Cayley graph?

Cayley Graphs

1. Draw one vertex for every group element, generator or not. (And don’t forget the identity!)
2. For every generator aj, connect vertex g to gaj by a directed edge from g to gaj. Label this edge with the generator.
3. Repeat step 2 for every element (i.e. vertex) g∈G.

Is a graph transitive?

A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular.

How do you prove a graph is not Hamiltonian?

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.

Can a graph be Eulerian and Hamiltonian?

A path is Eulerian if every edge is traversed exactly once. Clearly, these conditions are not mutually exclusive for all graphs: if a simple connected graph G itself consists of a path (so exactly two vertices have degree 1 and all other vertices have degree 2), then that path is both Hamiltonian and Eulerian.

Are all 3-regular graphs Hamiltonian?

Tait conjectured that every cubic polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Tait’s conjecture, the 46-vertex Tutte graph, in 1946. In 1971, Tutte conjectured that all bicubic graphs are Hamiltonian.

Is a regular graph connected?

In a complete graph of N vertices, each vertex is connected to all (N-1) remaining vertices. So, degree of each vertex is (N-1). So the graph is (N-1) Regular. For a K Regular graph, if K is odd, then the number of vertices of the graph must be even.

What is not a Hamiltonian graph?

A nonhamiltonian graph is a graph that is not Hamiltonian.

What is Hamiltonian graph example?

Hamiltonian graph – A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once.