... If G itself is connected, there is nothing more to prove. ... Since we have assumed that G is disconnected, we must now prove that G is connected. ...

... We can prove it by contradiction. Suppose that G is a 2-connected graph containing a vertex v that is adjacent to at least three vertices of degree 2, choose ...

... constructive proof of sufficiency by building an Euler trail T. Let T be empty in the start. Choose an arbitrary vertex 'x' in D. Since 'D' is connected, there ...

... subspace of X is connected. 4. Let X be a set and F is separating collection of functions f: X --> Y_f, each from X into a topological space Y_f. Prove that X ...

... thus M is not connected. We get a contradiction. Therefore, there must be some m ∈ M , such that g ( m ) = f ( m) − x = 0 ⇒ f ( m) = x (b) Proof: We want ...

... that are connected to f(v) in H, which is another way of saying that the degree of v in G is equal to the degree of f(v) in H. (c) Prove that isomorphic graphs ...

... PROBLEM (Exercise 2.29). Prove that X is connected if and only if X cannot be written as a union of two on-empty disjoint sets which are open relative to X. ...