On induced subgraphs of the cube

Abstract

Consider the usual graph Qⁿ defined by the n-dimensional cube (having 2ⁿ vertices and n2^{n − 1} edges). We prove that if G is an induced subgraph of Qⁿ with more than 2^{n − 1} vertices then the maximum degree in G is at least ($\text{1}\text{2}\text{− o(1))}\text{log}\text{n}$. On the other hand, we construct an example which shows that this is not true for maximum degree larger than.