Cycle Systems

Abstract

In this paper we show that the composition (symmetric difference) ofcycles is well-defined. So, such a collection {. . . , Ci, Ck, . . .} of cycles with a com-position operator,◦, is a matroid. As such, it has sets of independent, or basis,cycles that determine its rankr. This paper is concerned with independent anddependent sets of cycles within a cycle system. In particular, we enumerate thenumber of all possible basis sets in any cycle system of rankr≤6. Then we usea generating function to establish that the ratio of basis sets to all possiblerelementsets approachesc,0.287< c <0.289.