Colorings of Hamming-Distance Graphs

• Main Author: Harney, Isaiah H.
• Format: Others
• Published: UKnowledge 2017
• Subjects: Hamming distance; graphs; coloring; q-ary block codes; Algebra; Discrete Mathematics and Combinatorics
• Online Access: http://uknowledge.uky.edu/math_etds/49; http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1048&context=math_etds

Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into subgraph structures whose identification may be useful in determining the chromatic number.