Maximum density of vertex-induced perfect cycles and paths in the hypercube

Abstract

Let H and K be subsets of the vertex set V(Qd) of the d-cube Qd (we call H and K configurations in Qd). We say K is an exact copy of H if there is an automorphism of Qd which sends H to K. If d is a positive integer and H is a configuration in Qd, we define λ(H,d) to be the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, and make conjectures about λ(C2d,d) and λ(Pd+1,d) for larger values of d. In our proofs there are connections with counting the number of sequences with certain properties and with the inducibility of certain small graphs. In particular, we needed to determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.