# BookRiff

If you don’t like to read, you haven’t found the right book

## Which graph will have a Hamiltonian circuit?

A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.

## How do you determine if a graph has a Hamiltonian circuit?

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.

## Can a bipartite graph be Hamiltonian?

Let G=(A∣B,E) be a bipartite graph. To be Hamiltonian, a graph G needs to have a Hamilton cycle: that is, one which goes through all the vertices of G. As each edge in G connects a vertex in A with a vertex in B, any cycle alternately passes through a vertex in A then a vertex in B. Hence C can not be a Hamilton cycle.

## Is a Eulerian graph Hamiltonian?

A Hamiltonian circuit ends up at the vertex from where it started. Important: An Eulerian circuit traverses every edge in a graph exactly once, but may repeat vertices, while a Hamiltonian circuit visits each vertex in a graph exactly once but may repeat edges.

## Is K3 4 a Hamiltonian?

Notice also that the closures of K3,3 and K4,4 are the corresponding complete graphs, so they are Hamiltonian.

## How many Hamilton circuits are in K5?

For K5 count the number of distinct Eulerian circuits. K5 has 5!/(5*2) = 12 distinct Hamiltonian cycles, since every permutation of the 5 vertices determines a Hamiltonian cycle, but each cycle is counted 10 times due to symmetry (5 possible starting points * 2 directions).

## Can bipartite graph have odd vertices?

It is obvious that if a graph has an odd length cycle then it cannot be Bipartite. In Bipartite graph there are two sets of vertices such that no vertex in a set is connected with any other vertex of the same set).

## Can a graph have a Hamiltonian circuit but not an Euler circuit?

Take a graph which is just a cycle on at least 4 vertices, then add an edge between one pair of vertices. Where you added the edge, you will have an odd degree, so the graph cannot have an Eulerian cycle. But the original cycle gives a Hamiltonian cycle.

## Can a graph have a Euler circuit and a Hamiltonian circuit?

A circuit is any path in the graph which begins and ends at the same vertex. The whole subject of graph theory started with Euler and the famous Konisberg Bridge Problem. An Eulerian circuit passes along each edge once and only once, and a Hamiltonian circuit visits each vertex once and only once.

## Is the Petersen graph Hamiltonian and Eulerian justify?

There exists a hamiltonian path (see 2b), but no hamiltonian cycle. Thus, the Petersen graph is not hamiltonian. However, it is interesting to note that by deleting any vertex in the Petersen graph, it makes it hamiltonian. eulerian trail: a trail that contains every edge of the graph.

## How is a Hamiltonian circuit notated on a graph?

One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once; it does not need to use every edge. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA.

## How did the Hamiltonian circuit get its name?

Hamiltonian circuits are named for William Rowan Hamilton who studied them in the 1800’s. One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once; it does not need to use every edge.

## How is a Hamilton Circuit different from a Hamilton path?

Hamilton Path is a path that contains each vertex of a graph exactly once. Hamilton Circuit is a circuit that begins at some vertex and goes through every vertex exactly once to return to the starting vertex. Some books call these Hamiltonian Paths and Hamiltonian Circuits.

## How to find the lowest cost Hamiltonian circuit?

To answer this question of how to find the lowest cost Hamiltonian circuit, we will consider some possible approaches. The first option that might come to mind is to just try all different possible circuits. 1. List all possible Hamiltonian circuits 2. Find the length of each circuit by adding the edge weights 3.