Proofs of Two Minimum Circuit Cover Conjectures

LetGbe a 2-edge-connected graph withmedges andnvertices. The following two conjectures are proved in this paper. (i) The edges ofGcan be covered by circuits of total length at mostm+n−1. (ii) The vertices ofGcan be covered by circuits of total length at most 2(n−1), wheren⩾2.