Christian Lawson-Perfect and Colin Wright have set up an instance of Mastodon—a decentralized, open-source Twitter clone—as a place for mathy folks to be social. It’s appropriately named mathstodon.xyz, and because it’s open-source they were able to easily hack in support. Neato! I’ve never been on Twitter—the costs of the resulting distraction would far outweight any benefits for me—but mathstodon.xyz seems like it could be a fun place to discuss math without being endlessly distracting (we’ll see), so I decided to try it out for now: I’m @byorgey.

Here’s my initial entry to the #proofinatoot contest—the idea is to write a proof that fits in Mastodon’s 500-character limit for “toots” (you know, like a tweet, but more mastodon-y). To fit this proof into 500 characters I had to leave out a lot of details; it was a fun exercise to take a cool proof and try to distill it down to just its core ideas. Can you fill in the details I omitted? (Also, can you figure out what word is commonly used to refer to graphs with these properties?)

Let be a graph with . Any two of the following imply the third: 1. is connected; 2. is acyclic; 3. has edges.

: by induction. Any walk must reach a leaf. Delete it and apply the IH.

: by induction. Sum of degrees is , so there are at least two leaves. Delete one and apply the IH.

: Let have connected components. Since for each, the total number of edges is , hence .