Can you draw these graphs in one stroke?
Drawing a graph in one stroke means using every edge exactly once, without lifting your pencil and without retracing an edge. Rather than beginning with a rule, try the routes first and see what you notice.
What does “one stroke” mean?
Imagine drawing the graph below without lifting your pencil. You may pass through a vertex more than once, but every edge must be used exactly once.
Before looking for a rule, try some examples. If a route fails, notice where you became stuck and which edges were left unused.
Can you find the one-stroke route?
Use every edge exactly once, without lifting your pencil and without retracing an edge. There are four graphs to try, getting progressively harder.
One-stroke drawings are Eulerian paths
When can a graph be drawn in one stroke?
odd vertices
✓ One stroke
Start and finish at the same vertex.
odd vertices
✓ One stroke
Start at one odd vertex and finish at the other.
odd vertices
✕ No one-stroke route using every edge exactly once.
So before trying route after route, count the degrees. A graph with exactly two odd vertices must start at one and finish at the other.
Does a crossing count as a vertex?
Not necessarily. In a graph drawing, two edges can cross without meeting. A crossing counts as a vertex only when the graph tells you it is one — usually by marking it with a dot.
Compare this with the ★ graph above: there the star is a real vertex, so the four edges meeting it genuinely connect there. In the crossed square below, the centre is only a crossing.
There is no dot at the crossing, so it is not a vertex.
Repair the graph — add or remove one edge
Each graph is not currently drawable in one stroke. Make exactly one change, then prove your repair by tracing the route. The first two teach the two kinds of repair; the later graphs make you decide what to do.
What have we learned?
One-stroke drawing puzzles are Euler-trail problems. The useful question is not “Which route should I try first?” but “How many odd-degree vertices does this graph have?”
Zero odd vertices: possible, finishing where you started.
Two odd vertices: possible, starting at one odd vertex and finishing at the other.
More than two odd vertices: impossible until the graph is changed.
Adding or removing one edge changes the degrees of exactly two vertices. That gives you a systematic way to repair a graph instead of guessing.
Download the printable lesson materials and supplementary examples for Drawing Without Lifting Your Pencil.
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