New bounds on a hypercube coloring problem

Authors:
Hung Quang Ngo;Ding-Zhu Du;Ronald L. Graham
Affiliations:
Department of Computer Science and Engineering, State University of New York, Buffalo, NY;Department of Computer Science and Engineering, University of Minnesota, 200 Union street, EE/CS Building, room 4-192, Minneapolis, MN;Department of Computer Science, University of California at San Diego, La Jolla, CA
Venue:
Information Processing Letters
Year:
2002

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Abstract

In studying the scalability of optical networks, one problem which arises involves coloring the vertices of the n-cube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of χ??(n), the minimum number of colors needed, appears to be a difficult problem. In this note, we improve the known lower and upper bounds of χ??(n) and indicate the connection of this coloring problem to linear codes.