Covering the edges of a graph by a prescribed tree with minimum Raphael Yuster Summary: Covering the edges of a graph by a prescribed tree with minimum overlap Noga Alon Yair Caro Raphael Yuster Abstract Let H = (VH, EH) be a graph, and let k be a positive integer. A graph G = (VG, EG) is H-coverable with overlap k if there is a covering of the edges of G by copies of H such that no edge of G is covered more than k times. Denote by overlap(H, G) the minimum k for which G is H-coverable with overlap k. The redundancy of a covering that uses t copies of H is (t|EH| - |EG|)/|EG|. Our main result is the following: If H is a tree on h vertices and G is a graph with minimum degree (G) (2h)10 + C, where C is an absolute constant, then overlap(H, G) 2. Furthermore, one can find such a covering with overlap 2 and redundancy at most 1.5/(G)0.1 . This result is tight in the sense that for every tree H on h 4 vertices and for every function f, the problem of deciding if a graph with (G) f(h) has overlap(H, G) = 1 is NP-Complete. 1 Introduction All graphs considered here are finite, undirected and simple, unless otherwise noted. For the Collections: Mathematics