On the structure of run-maximal strings

We present a combinatorial structure consisting of a special cover of a string by squares. We characterize the covering property of run-maximal strings, i.e. strings achieving the maximal number of runs. The covering property leads to a compression scheme which is particularly efficient for run-maximal strings. It also yields a significant speed improvement in the computer search for good run-maximal string candidates. The implementation of the search and preliminary computational results are discussed.