Packing and Covering Balls in Graphs Excluding a Minor

Packing and Covering Dense Graphs

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 o...

Packing and covering triangles in graphs

Packing and Covering Triangles in Planar Graphs

Tuza conjectured that if a simple graph G does not contain more than k pairwise edgedisjoint triangles, then there exists a set of at most 2k edges that meets all triangles in G. It has been shown that this conjecture is true for planar graphs and the bound is sharp. In this paper, we characterize the set of extremal planar graphs.

Packing and covering δ-hyperbolic spaces by balls

We consider the problem of covering and packing subsets of δ-hyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic graph G and a positive number R, let γ(S, R) be the minimum number of balls of radius R covering S. It is kno...

Packing and covering with balls on Busemann surfaces

In this note we prove that for any compact subset S of a Busemann surface (S, d) (in particular, for any simple polygon with geodesic metric) and any positive number δ, the minimum number of closed balls of radius δ with centers at S and covering the set S is at most 19 times the maximum number of disjoint closed balls of radius δ centered at points of S: ν(S) ≤ ρ(S) ≤ 19ν(S), where ρ(S) and ν(...