Extensions of matroid covering and packing

Let M be a loopless matroid on E with rank function r(m). Let beta(M) = max(empty set not equal X subset of E) vertical bar X vertical bar/R-M(X) and phi(M) = Minr(M)(X)<r(M) vertical bar E vertical bar-vertical bar X vertical bar/r(M)(E)-r(M)(X). The Matroid Covering and Packing Theorems state that the minimum number of independent sets whose union is E is [beta(M)], and the maximum number of disjoint bases is [phi(M)]. We prove refinements of these theorems. If beta(M) = k + epsilon, where k >= 0 is an integer and 0 <= epsilon < 1, then E can be partitioned into k + 1 independent sets with one of size at most epsilon . r(M)(E). If ca(M) = k + epsilon, then M has k + 1 disjoint independent sets such that k are bases and the other has size at least epsilon . r(M)(E). We apply these results to truncations of cycle matroids to refine graph-theoretic results due to Chen, Koh, and Peng in 1993 and to Chen and Lai in 1996. (C) 2018 Elsevier Ltd. All rights reserved.