A Radio Coloring of a Hypercube

A radio coloring of a graph G is an assignment of nonnegative integers to its nodes so that each pair of adjacent nodes have color numbers that differ by at least two, and any pair of nodes at distance 2 have different colors. Every graph has a radio coloring by simply assigning the odd integers 1, 3, 5, ......, but there is then a big difference between the smallest and largest colors. We define the span of a radio coloring of G as one plus the difference between the smallest and largest colors. We study radio colorings of a hypercube with the objective of finding such a coloring with minimum span. We develop a formulation for what we believe is the complete solution to this question in the form of a conjecture.