The chromatic number of the square of the $8$-cube

Janne I. Kokkala, Patric R. J. Östergård
2017 Mathematics of Computation  
A cube-like graph is a Cayley graph for the elementary abelian group of order 2 n . In studies of the chromatic number of cube-like graphs, the kth power of the n-dimensional hypercube, Q k n , is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph Q k n can be constructed with one vertex for each binary word of length n and edges between vertices exactly when the Hamming distance between the corresponding words is at most k.
more » ... y, a proper coloring of Q k n corresponds to a partition of the n-dimensional binary Hamming space into codes with minimum distance at least k + 1. The smallest open case, the chromatic number of Q 2 8 , is here settled by finding a 13-coloring. Such 13-colorings with specific symmetries are further classified. * MSC Classification: Primary 05C15, Secondary 94B25
doi:10.1090/mcom/3291 fatcat:6afwfvzw2vdmfn2ovdcpvilzta