Number of proper paths in edge-colored hypercubes

Given an integer 1 <= j < n, define the (j)-coloring of a n-dimensional hypercube H-n to be the 2-coloring of the edges of H-n in which all edges in dimension i, i <= i <= j, 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. (C) 2018 Elsevier Inc. All rights reserved.