My answer:
My answer:
My answer:
My answer:
My answer:
Imagine the graph is a huge map, and you can’t just try every possibility to see if it is true that we will always come back if all of the vertices have even degrees. What are you going to do?
Try it with parts of the map, and then step by step to enlarge the partials. If we can prove the conclusion works in partials, then we can say that it will works in larger partials too. And in the end, the whole map works.
Let’s start with a single vertex, the first job is to only traverse all of the edges it connected with. Then, to enlarge the map, we start with the vertex that we’ve already covered and still has available edges but only focus on traversing all of the left edges. Eventually, we will cover the whole map by parts. The next step is to try to splice two parts in once, then three parts in once. Seem like, we can use this strategy to cover the whole map in once. In the end, we proved the Euler Circuit Theorem, which says that if you have a connected graph where every vertex has even degree, then it’s possible to start at any vertex, traverse all of the edges in the graph, never going over the same edge twice, and returning to where we started.
Now, let’s review the whole question.
First, we abstracted the map to points and lines, just tried to understand the question deeply.
Then, we did several attempts, but always got stuck somewhere. Even we are frustrated but also got some clues. We found out all of the points connected with odd lines. To get better illustrating, we come out some concepts, like vertex, edge and the degree of the vertex. If there are some vertices has odd degree, we can’t transverse all the edges.
Last, to get a better understanding of this problem, we raised two questions. The first one is some of the vertices have odd degree, some don’t. The last one is all of the vertices have even degree. In the process of figuring out the questions.
We learned: