On the Maximum Number of Maximum Independent Sets

Abstract

We give a very short and simple proof of Zykov’s generalization of Turan’s theorem, which implies that the number of maximum independent sets of a graph of order n and independence number $$\alpha $$ with $$\alpha <n$$ is at most $$\left\lceil \frac{n}{\alpha }\right\rceil ^{n\,\mathrm{mod}\,\alpha } \left\lfloor \frac{n}{\alpha }\right\rfloor ^{\alpha -(n\,\mathrm{mod}\,\alpha )}$$ . Generalizing a result of Zito, we show that the number of maximum independent sets of a tree of order n and independence number $$\alpha $$ is at most $$2^{n-\alpha -1}+1$$ , if $$2\alpha =n$$ , and, $$2^{n-\alpha -1}$$ , if $$2\alpha >n$$ , and we also characterize the extremal graphs. Finally, we show that the number of maximum independent sets of a subcubic tree of order n and independence number $$\alpha $$ is at most $$\left( \frac{1+\sqrt{5}}{2}\right) ^{2n-3\alpha +1}$$ , and we provide more precise results for extremal values of $$\alpha $$ .