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.

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 2 5 3 6 2

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.”

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.

The Brute force algorithm Example: Circuit Weight ABCDA 4+13+8+1 = 26 ABDCA 4+9+8+2 = 23 ACBDA 2+13+9+1 = 25 13

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 1+8+13+4=26

Sorted Edges algorithm 13

Sorted Edges algorithm AD

Sorted Edges algorithm AD AC

Sorted Edges algorithm AD AC BD

Sorted Edges algorithm AD AC BD BC