Inventive graphs

When Torrin and I were playing with wooden railways, Torrin spontaneously noticed "this track has got four holes"

– it wasn't actually this track, but that doesn't matter – I thought, oh good, T both is interested in counting and has noticed that a track defines holes. And I grabbed some scrap paper and drew a picture of "junctions", "edges", and "holes". (The top drawing on the next sheet.) I counted with Torrin how many junctions it has, how many edges, and how many holes. Then I encouraged Torrin (who needed little encouragement) to draw another picture of junctions and edges, so that we could count the junctions, edges and holes. We rapidly had three more to count.

Torrin was keen to make more pictures. I didn't ever suggest that he should make pictures of the most diverse types possible; he spontaneously did so, making a big circular track with lots of identical bubbles around the edge, all the junctions having 3 or 4 edges coming out of them; and making another big circular track with a megajunction in the centre and lots of minimal three-edged junctions around the periphery. And then one in which the track made loops so there were several junctions with edges that connected directly to the same junction.

For every drawing, we counted the junctions, edges, and holes, and at some point I started adding together the number of holes and the number of junctions, and then pointed out a pattern that this sum was always one bigger than the number of edges. And Torrin kept going and made drawings of a track with no holes – of a track where all the junctions had two edges and there was only one hole; and then even (secretively and with a big smile) one with no holes and no edges! (just a single dot).

I don't like being a pushy parent, nor do I like bragging, but I am really impressed how inventive Torrin was - making all sorts of extreme graphs, including several pathological cases where I feared the pattern we were testing (Euler's theorem) might break down.
[I don't think Torrin actually understood the pattern, because I don't think he yet really understands adding one and taking away one (except where small numbers of M&Ms are involved) and he definitely isn't robustly familiar yet with larger numbers such as 32.)