Saturated Subgraphs of the Hypercube
We say a graph is (Qn,Qm)-saturatedif it is a maximalQm-free subgraph of then-dimensional hypercubeQn. A graph is said to be (Qn,Qm)-semi-saturatedif it is a subgraph ofQnand adding any edge forms a new copy ofQm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectivelys-sat(Qn,Qm)). We prove that$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$for fixedm, disproving a conjecture of Santolupo that, whenm=2, this limit is 1/4. Further, we show by a different method that sat(Qn,Q2)=O(2n), and thats-sat(Qn,Qm)=O(2n), for fixedm. We also prove the lower bound$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$thus determining sat(Qn,Q2) to within a constant factor, and discuss some further questions.