Stijn Cambie, The 69-conjecture and more surprises on the number of independent sets

Various types of independent sets have been studied for decades. As an example, the minimum number of maximal independent sets in a connected graph of given order is easy to determine (hint; the answer is written in the stars). When considering this question for twin-free graphs, it becomes less trivial and one discovers some surprising behaviour. The minimum number of maximal independent sets turns out to be;

• logarithmic in the number of vertices for arbitrary graphs,
• linear for bipartite graphs
• and exponential for trees.

Finally, we also have a sneak peek on the 69-conjecture, part of an unpublished work on an inverse problem on the number of independent sets.

In this talk, we will focus on the basic concepts, the intuition behind the statements and sketch some proof ideas.

The talk is based on joint work with Stephan Wagner, with the main chunk being available at arXiv:2211.04357.