Collaborative mathematics tends to be extremely fruitful. This is epitomised by the recent ‘polymath’ projects, culminating in a nice proof of density Hales-Jewett along with the frighteningly impressive results on bounded gaps between primes. On a much smaller scale, several of us were contemplating a bunch of interesting and completely unrelated problems we devised in the downstairs kitchen of a converted house in Burrell’s Field, Trinity College, Cambridge.
We’ve made non-trivial progress on many of these, and even succeeded in solving one! It was a natural question, which grew out of a much easier problem on an IMO shortlist:
Suppose you begin with a finite graph, and are allowed to apply operations of the following form:
- Deletion: Choose a vertex of odd degree and delete it (along with all edges incident with it).
- Duplication: Produce an identical copy of the graph, and connect each vertex in the original graph to its…
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