Covering the Edges of a Graph by a Prescribed Tree with Minimum Overlap

LetH=(VH, EH) be a graph, and letkbe a positive integer. A graphG=(VG, EG) isH-coverable with overlap kif there is a covering of the edges ofGby copies ofHsuch that no edge ofGis covered more thanktimes. Denote by overlap(H, G) the minimumkfor whichGisH-coverable with overlapk. Theredundancyof a covering that usestcopies ofHis (t|EH|−|EG|)/|EG|. Our main result is the following: IfHis a tree onhvertices andGis a graph with minimum degreeδ(G)⩾(2h)10+C, whereCis an absolute constant, then overlap(H, G)⩽2. Furthermore, one can find such a covering with overlap 2 and redundancy at most 1.5/δ(G)0.1. This result is tight in the sense that for every treeHonh⩾4 vertices and for every functionf, the problem of deciding if a graph withδ(G)⩾f(h) has overlap(H, G)=1 is NP-complete.