Sphere Packing Proper Colorings of an Expander Graph

Abstract

We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^{n}$ where a code is a set of proper q-colorings of some fixed n-vertex graph G. We then say that a set of M proper q-colorings of G form a $(G, M, d)$ code if any pair of colorings in the set have Hamming distance at least d. This directly generalizes typical $(n, M, d)$ codes of q-ary strings of length n since we can take G as the empty graph on n vertices. We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed $(\delta, \lambda) \in [{0, 1}] \times [-1, 1]$ and positive integer d, let $f_{\delta, \lambda, d}(n)$ denote the maximum M such that there exists some d-regular graph G on at most n vertices with normalized second eigenvalue at most $\lambda $ that has a $(G, M, d)$ code. We study the growth of f as n goes to infinity. We partially characterize the regimes of $(\delta, \lambda)$ where f grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes.