TY - JOUR

T1 - Packing and covering dense graphs

AU - Alon, Noga

AU - Caro, Yair

AU - Yuster, Raphael

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1998

Y1 - 1998

N2 - Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants ∈(H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 - ∈(H))|G|, then the H-packing number of G and the H-covering number of G can be computed in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given.

AB - Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants ∈(H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 - ∈(H))|G|, then the H-packing number of G and the H-covering number of G can be computed in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given.

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U2 - 10.1002/(SICI)1520-6610(1998)6:6<451::AID-JCD6>3.0.CO;2-E

DO - 10.1002/(SICI)1520-6610(1998)6:6<451::AID-JCD6>3.0.CO;2-E

M3 - Article

AN - SCOPUS:27844464109

VL - 6

SP - 451

EP - 472

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

IS - 6

ER -