Eulerian Graph: An Eulerian graph is a graph that can be decomposed into a disjoint union of two disjoint sets, E and F, such that every edge of the graph c...
Eulerian Graph:
An Eulerian graph is a graph that can be decomposed into a disjoint union of two disjoint sets, E and F, such that every edge of the graph connects a vertex in E to a vertex in F. In simpler terms, an Eulerian graph can be divided into two parts, where any two vertices in the two parts are connected by exactly one edge.
Hamiltonian Graph:
A Hamiltonian graph is an Eulerian graph that is also a Hamiltonian graph. This means that it can be mapped onto a Hamiltonian system, meaning that the energy of a particle in the system is determined by the positions of the vertices in the graph.
Key Differences:
Eulerian graphs can be decomposed into a finite number of disjoint sets, while Hamiltonian graphs need to be decomposed into a finite number of disjoint sets.
Eulerian graphs are always planar, while Hamiltonian graphs can be both planar and non-planar.
Eulerian graphs have a Euler characteristic, which is a numerical value that can be used to determine whether a graph is Eulerian. The Euler characteristic of a Hamiltonian graph is equal to 1, which means that it is Hamiltonian