On the Number of Maximal Independent Sets in a Graph

We show that the number of maximal independent sets of size exactly k in any graph of size n is at most [ n/k ]k-(n mod k) ([ n/k ] +1)n mod k. For maximal independent sets of size at most k the same bound holds for k <= n/3. For k > n/3 a bound of approximately 3n/3 is given. All the bounds are exactly tight and improve Eppstein (2001) who give the bound 3⁴ᵏ⁻ⁿ4ⁿ⁻³ᵏ on the number of maximal independent sets of size at most k, which is the same for n/4 <= k <= n/3, but larger otherwise. We give an algorithm listing the maximal independent sets in a graph in time proportional to these bounds (ignoring a polynomial factor), and we use this algorithm to construct algorithms for 4- and 5- colouring running in time O(1.7504^n) and O(2.1593^n), respectively.