4. In this problem we will complete steps to show Theorem 5.6.3 in LLM: . A connected graph has an (a) Give a direct proof of the following: If a connected graph has an Euler tour, the every vertex has (b) Now we step through showing that if every vertex has even degree, then the connected graph has Euler tour if and only if every vertex has even degree. even degree. an Euler tour. Let's consider the longest walk in the graph that does not traverse any edge more than once, L i. Why must L exist? ii. Why must L be a closed walk? ii. Why must L include all edges incident to the starting/ending vertex? iv. Prove by contradication the L must be Eulerian, i.e. L must include all the edges in the graph. (Why does this complete the proof?)