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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
Noga Alon
Yair Caro
Raphael Yuster
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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics