1. Euler circuit
Theorem 1 If a graph G has an Eulerian path, then it must have exactly two odd vertices. Theorem 2 If a graph G has an Eulerian circuit, then all of its vertices must be even vertices.

2. Euler circuit Hierholzer's algorithm: Create a circuit C 1 2 3 4 5 6 1
Delete edges from the graph
Find a vertex v  C, v  C1 and create new circuit

3. Hamiltonian Circuit
If there is a vertex of degree one in a graph then it is impossible for it to have a Hamiltonian circuit.
Dirac’s Theorem- “If G is a simple graph with n vertices where n>==3 such that the degree of every vertex in G is at least n/2, then G has a Hamiltonian circuit.”
Ore’s Theorem- “If G is a simple graph with n vertices with n>=3 such that the sum of degrees for every pair of non-adjacent vertices is greater than n, then G has a Hamiltonian circuit.”

4. Hamiltonian Circuit The Brute force algorithm:
1.     List all possible Hamiltonian circuits
2.     Find the length of each circuit by adding the edge weights
3.     Select the circuit with minimal total weight.

5. The Brute force algorithm
Example:
Circuit
Weight
ABCDA
= 26
ABDCA
= 23
ACBDA
= 25 13

6. Nearest Neighbor Algorithm (NNA)
1.     Select a starting point.
2.     Move to the nearest unvisited vertex (the edge with smallest weight).
3.     Repeat until the circuit is complete. 13
ADCBA with a total weight of =26

7. Sorted Edges algorithm13

8. Sorted Edges algorithmAD

9. Sorted Edges algorithmAD AC

10. Sorted Edges algorithmAD AC BD

11. Sorted Edges algorithmAD AC BD BC