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Question

If our graph has isolated vertices then it can have euler circuit. But with isolated vertices we can never have hamiltonian circuit.

so we can have euler circuit even in a disconnected graph but hamiltonian circuit is possible always on a connected graph.

Am I right?

Answer 1

An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For finite connected graphs the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the weaker sense if and only if each connected component has an Eulerian cycle.

For Hamiltonian graph there should exist a circuit connecting all the vertices of graph, so isolation it's not possible

So we can conclude that Euler circuit is possible of disconnected graph, but not same for Hamiltonian circuit