On the number of proper paths between vertices in edge-colored hypercubes

Abstract

Given an integer $1\le j<n$, 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 $i$, $1\le i\le j$, have color $1$ and all other edges have color $2$. Cheng et al. [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] 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, $j\ge 2$. In this note, we determine the number of different shortest proper paths in $(j)$-colored hypercubes for arbitrary $j$.

9 pages