Number of proper paths in edge-colored hypercubes

Given an integer 1 ≤ j < n, define the (j)-coloring of a n-dimensional hypercube Hn to be the 2-coloring of the edges of Hn in which all edges in dimension i, 1 ≤ i ≤ j, have color 1 and all other edges have color 2. Cheng et al. (2017) determined the number of distinct shortest properly colored paths between a pair of vertices for the (1)-colored hypercubes. It is natural to consider the number for (j)-coloring, j ≥ 2. In this note, we determine the number of different shortest proper paths in (j)-colored hypercubes for arbitrary j. Moreover, we obtain a more general result.