Subgraph Coverings and Edge Switchings

By using the so-defined circuit/path transformations together with an edge-switching method, the following conjectures are proved in this paper. (i) The edges of a connected graph on n vertices can be covered by at most ⌈n2⌉ paths, which was conjectured by Chung. (ii) The edges of a 2-connected graph on n vertices can be covered by at most ⌊2n−13⌋ circuits, which was conjectured by Bondy. An immediate consequence of (ii) is a theorem of Pyber that the edges of a graph on n vertices can be covered by at most n−1 edges and circuits, which was conjectured by Erdös, Goodman, and Pósa.