TY - JOUR

T1 - Covering the edges of a graph by a prescribed tree with minimum overlap

AU - Alon, Noga

AU - Caro, Yair

AU - Yuster, Raphael

N1 - Funding Information:
* Research supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences.

PY - 1997/11

Y1 - 1997/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031280922&partnerID=8YFLogxK

U2 - 10.1006/jctb.1997.1768

DO - 10.1006/jctb.1997.1768

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AN - SCOPUS:0031280922

VL - 71

SP - 144

EP - 161

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -