task 6

QUESTION 1: (DEFINITION)

EULER PATH

An Euler path is a path that uses every edge of a graph 

exactly once

An Eulerian trail (or Eulerian path) is a trail in a graph which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycleis an Eulerian trail which starts and ends on the same vertex. 

Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit.


EULER CIRCUIT

An Euler circuit is a circuit that uses every edge of a graph 

exactly once.

Every vertex of this graph has an even degree, therefore this is an Eulerian graph.

 Following the edges in alphabetical order gives an Eulerian circuit/cycle.

QUESTION 2.
HOW TO DIFFERENTIATE BETWEEN EULER PATH AND EULER CIRCUIT?

ANSWER:

Euler Path	Euler Circuit
A path that uses every edges of a graph exactly once	A circuit that uses every edges of a graph exactly once
Start and end at different vertices	Start and end at same vertex
If a graph has Euler path, then it must have exactly 2 odd vertices	If a graph has Euler circuit, then all of its vertices must be even vertices

HAMILTON PATH

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian Path can start from one point and ends up at another point.




 HAMILTON CYCLE

A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Hamiltonian Cycle starts from one point and returns to the starting point after traveling it's path.














QUESTION 5
HOW TO DIFFERENTIATE BETWEEN HAMILTON PATH AND HAMILTON CIRCUIT?

ANSWER:

Hamilton Path	Hamilton Circuit
A path that passed every vertex exactly once	A simple circuit that passed every vertex and return to the starting vertex
There is a vertex that has one degree	There is a 2 or more than one degree in the vertex