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