TITLE: Is it true that a graph with many edges has a long route?
QUESTION [5 upvotes]: Is the following sentence true or not?
If we have a graph with $n$ vertices, and $e$ edges, if $e > 100 n$, then we always have a $100$-long route in the graph.
I think it is true, I tried to use that $2e = \sum_{i=1}^{n}d_i$, which means that the average degree for a vertex is at least $200$. Can't really go forward. Any help? :)
Edit: Answer below.

REPLY [4 votes]: Today I got the answer:
The average degree is $200$, if all of them are over $100$, I am done, because I can just use one of them as starter: $v_1$ and this vertex has at least $99$ neighbours. Choose one of them(not $v_1$), let it be $v_2$, this one has at least $98$, and so on, in this way, I get my $100$-long route.
If some of the degrees are lower than $100$, delete them from the graph until all of the degrees are over $100$. The graph won't get empty since the average is $200$. I can use what I proved above for the remaining graph.