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

TL;DR

This paper extends previous work on counting shortest properly colored paths in hypercubes by analyzing the case where multiple edge colors are used, providing a general formula for arbitrary colorings.

Contribution

It introduces a general method to determine the number of shortest proper paths in j-colored hypercubes for any j, expanding prior results limited to j=1.

Findings

Derived a formula for shortest proper paths in j-colored hypercubes

Extended previous results from 1-color to multiple colors

Provides combinatorial insights into hypercube colorings

Abstract

Given an integer $1\leq j <n$, 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 $i$, $1\leq i \leq j$, 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, $j\geq 2$. In this note, we determine the number of different shortest proper paths in $(j)$-colored hypercubes for arbitrary $j$.