Hamiltonian and Eulerian Paths/Cycles

Eulerian graph

"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." "Eulerian graph" is a graph containing an Eulerian cycle.

Examples of Graphs with Hamiltonian cycles

-A complete graph with more than two vertices is Hamiltonian V = n=7 E = n(n-1)/2 -Every cycle graph is Hamiltonian V = n=6 E = n=6

Hamiltonian path

a path in an undirected or directed graph that visits each vertex exactly once

Fleury's algorithm

finding Euler paths/cycles -at most two vertices of odd degree - stars at odd if any -choose E that will not disconnect the graph - E cycle if there are no odd -E path if exactly 2 odd

undirected E path

iff at most two vertices have odd degree, and if all of its vertices with nonzero degree belong to a single connected component. if 2 odd start at one and end at the other

undirected has E cycle

iff every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component

directed E cycle

iff every vertex has equal in degree and out degree, and all of its vertices with nonzero degree belong to a single strongly connected component. not only is there a path between every pair of vertices (perhaps only in one direction), but there exists a path from every vertex to every other vertex in the graph

Hamiltonian graph

is just a graph that contains a Hamiltonian cycle.

existence of Eulerian

it is necessary that no more than two vertices have an odd degree

Hamiltonian cycle

a Hamiltonian path that is a cycle." (The first and last vertices are the same)

Eulerian Path

a trail in a graph which visits every edge exactly once

Eulerian cycle/circuit

an Eulerian trail which starts and ends on the same vertex If there are no vertices of odd degree, all Eulerian trails are cycles

V = n How many Hamiltonian cycles/paths?

n! NP complete = no known polynomial-time algorithm