Eulerian trail (or Eulerian path)

Question

I found this question in one oldie book

It is possible to draw figure A below without lifting your pencil in such a way that you never draw the same line twice. However, no matter how hard you try, it seems impossible to draw Figure B in this way. Can you find criteria that will allow you to quickly determine whether any given figure can or cannot be drawn in this way?

I read one or two answers for this very website. However, as a beginner I hardly understood the graph theory parts. I used common sense to conclude Figure A follows, but Figure B doesn't.

I want help in the part of the criteria of solving and its proof too.

Answer 1

In view of your question, I will try to answer without using any graph theory terminology. Any time you visit an intersection of lines, you must use two of the lines - one to enter and a different one to leave - unless the intersection is the start of your path (in which you don't use a line to get there, you just use one to leave) or the end of your path (similarly).

Therefore, every intersection must have an even number of lines attached to it, except for the beginning and end intersection of your path which will have on odd number (but see final comment). Your path can only have one beginning and one end, so if it is possible to traverse the diagram in a single path, there must be only two "odd intersections". In the right-hand diagram there are four, so it is impossible.

Comment. It might be that your path begins and ends at the same intersection, in which case this intersection will also be "even".

The complete answer to the problem: assuming your lines do not fall into two or more separate "islands" which are totally disconnected from each other, then there will be a path covering all lines without repetition, provided there are at most two "odd intersections". There will be such a path which returns to its starting point if there are no "odd intersections" at all.