a. Prove that if a walk in a graph contains a repeated edge, then the walk contains a repeated vertex.
b. Explain how it follows from part (a) that any walk with no repeated vertex has no repeated edge.

(a)

To prove:

Prove that if a walk in a graph contains a repeated edge, then the walk contains a repeated vertex.

Given information:

a walk in a graph contains a repeated edge.

Proof:

A walk from v to w is a finite sequence of alternatively adjacent vertices and edges of a graph.

To proof: If a walk is a graph contains a repeated edge, then the walk contains a repeated vertex.

PROOF BY CASES:

Let G be a graph and let W be a walk in the graph G such that W contains some repeated edge e.

Let u and v be the endpoints of the edge e.

If u = v, then the edge e is a loop at vertex u is then a repeated vertex, which would complete the proof. Let us then assume that u≠v.

FIRST CASE: W contains two copies of uev or W contains two copies of veu

In this case both u and v are repeated vertices

(b)

Explain how it follows from part (a) that any walk with no repeated vertex has no repeated edge.