Cycle Cover Ratio of Regular Matroids

A cycle in a matroid is a disjoint union of circuits. This paper proves that every regular matroid M without coloops has a set S of cycles whose union is E(M) such that every element is in at most 3 of the cycles in S. It follows immediately from this that, on average, each element of M is in at most 3 members of the cycle cover S. This improves on a 1989 result of Jamshy and Tarsi who proved that there is a cycle cover for which this average is at most 4, and conjectured that a cycle cover exists for which the average is at most 2.