Can the following picture be drawn without lifting the pen from the paper or retracting?

Question

Related (essentially a duplicate, since the reasoning in the answers applies here): "Proof : cannot draw this figure without lifting the pen". See also, lots more in the "Related" list of this page's sidebar. — Blue, Jun 17 '19 at 07:38
@cmi: Correct! The path cannot be drawn. You should give your reasoning in an answer, so that we can upvote it. :) — Blue, Jun 17 '19 at 07:52

score 1 · Answer 1 · answered Jun 17 '19 at 11:23

1

Definitely not, as A,B,C,E all have an odd number of incident paths.

--- rk

answered Jun 17 '19 at 11:23

Dr. Richard Klitzing

5,757

score 1 · Answer 2 · answered Jun 19 '19 at 22:22

Imagine vertices as nodes of a graph and lines joining them as edges. Now, a graph can be drawn without lifting the pen from the paper or retracting if it has a Eulerian Path.

A graph has a Eulerian path if
1. Every node has even degree (even number of edges)
2. Exactly two nodes have odd degree and other's have even degree.

The degree is the count of edges passing through the vertex.

In your case, this fails, as this drawing has more than 2 vertices with odd degrees

Can the following picture be drawn without lifting the pen from the paper or retracting?

2 Answers2