if (dis)proving a conjecture on graph theory can be done just by a counter example then can every (dis)proof be mapped actually to a counter-example?
Your question isn't entirely clear, but it seems you're asking if every false graph property has a counterexample. Obviously this is true: by definition there must be some graph for which the property doesn't hold. On the other hand, a proof that the property doesn't hold for all graphs may not necessarily give an example of such a graph; this is called a non-constructive proof.