漢諾圖上的哈密頓路徑

Abstract

本文給出了 n 階 2 維漢諾圖（又稱漢諾塔圖、河內圖）上哈密頓路徑的數量，其漸進表現是 h(n) ∼ 25×16^n/624 。這類漢諾圖上的哈密頓路徑總數量與起點在最上面的顶點的哈密頓路徑數量的對數的比值漸進至 2。同時，當這類漢諾圖上三個方向的平行邊分別被 x, y, z 這三個數\n加權後，我們也推導出了它們的哈密頓路徑的加權和，其漸進表現為h′(n) ∼(25w*16^n(xyz)^(3n−1))/(16*27*13)其中 w =(x + y + z)^2/(xyz)。

We’ve derived the number of Hamiltonian walks on the two­dimensional Hanoi graph at stage n, whose asymptotic behaviour is given by h(n) ∼ 25×16^n/624 .\nAnd the asymptotic behaviour the logarithmic ratio of the number of Hamiltonian walks on these Hanoi graphs with that one end at the topmost vertex is given by 2. When the parallel edges in the three directions on these Hanoi graphs are weighted by three numbers, x, y, z, the weighted sum of their Hamiltonian paths is also derived by us, and the asymptotic behaviour of it is given by\nh′(n) ∼(25w*16^n(xyz)^(3n−1))/(16*27*13) in which w =(x + y + z)^2/(xyz).