Dimer coverings on the Tower of Hanoi graph

We present the number of dimer coverings N _d (n) on the Tower of Hanoi graph TH _d (n) at n stage with dimension 2 ≤d ≤ 5. When the number of vertices v(n) is even, N _d (n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as STH d=limn→∞ln N _d (n)/v(n), that can be shown exactly for TH ₂ , while its lower and upper bounds can be derived in terms of the results at a certain n for TH _d (n) with 3 ≤d ≤ 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of STH d with d=3 and 5 can be calculated with at least 100 correct digits.