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### A Nearly Linear Time Algorithm For The Half Integral Parity Disjoint Paths Packing Problem [chapter]

Ken-ichi Kawarabayashi, Bruce Reed
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
We consider the following problem, which is called the half integral parity disjoint paths packing problem.  ...  paths packing problem, i.e., without the parity requirement.  ...  parities of half-integral disjoint paths packing.  ...

### Experimental Study of Independent and Dominating Sets in Wireless Sensor Networks Using Graph Coloring Algorithms [chapter]

Dhia Mahjoub, David W. Matula
2009 Lecture Notes in Computer Science
Furthermore, if we relax the full domination constraint then we obtain a partitioning of the network into disjoint dominating and nearly-dominating sets of nearly equal size, providing better redundancy  ...  The disjoint maximal independent sets constitute a collection of disjoint dominating sets that offer good network coverage.  ...  Nearly-Equal Sized Independent Sets Collection of Disjoint Dominating Sets Node Packing in the Independent Sets We consider r=0.12 with several n values as depicted in Table 4 .  ...

### The Almost Intersection Property for Pairs of Matroids on Common Groundset

2020 Electronic Journal of Combinatorics
Theorem 4 . 2 . 42 If M and N are nearly finitary, then M ∨ N is a nearly finitary matroid. Theorem 4 . 3 . 43 If M and N are nearly finitary, then (M, N ) has the Packing/Covering Property. Proof.  ...  For the rest of this section assume that M and N are matroids on a common ground set E.A packing for (M, N ) is a pair (S, T ) of disjoint subsets of E such that cl M (S) = cl N (T ) = E.  ...

### Asymptotically Optimal Tree-Packings in Regular Graphs

Alexander Kelmans, Dhruv Mubayi, Benny Sudakov
2001 Electronic Journal of Combinatorics
Clearly, an $n$ vertex graph contains at most $n/t$ vertex disjoint trees isomorphic to $T$.  ...  that for every $\epsilon >0$, there exists a $D(\epsilon,t)>0$ such that, if $d>D(\epsilon,t)$ and $G$ is a simple $d$-regular graph on $n$ vertices, then $G$ contains at least $(1-\epsilon)n/t$ vertex disjoint  ...  Then G contains at least (1 − )n/t vertex disjoint copies of T . the electronic journal of combinatorics 8 (2001), #R38 A matching in H is a set of pairwise disjoint edges of H.  ...

### Page 4818 of Mathematical Reviews Vol. , Issue 83m [page]

1983 Mathematical Reviews
D. 83m:05044 Packing nearly disjoint sets. Combinatorica 2 (1982), no. 1, 91-97.  ...  The proof actually provides an algorithm for finding [|@|/n| pairwise disjoint sets in @.  ...

### Infinite matroid union [article]

Elad Aigner-Horev, Johannes Carmesin, Jan-Oliver Fröhlich
2012 arXiv   pre-print
We introduce a superclass of the finitary matroids, the nearly finitary matroids, and prove that the union of two nearly finitary matroids is a nearly finitary matroid.  ...  We then extend the base packing theorem for finite matroids to finite families of co-finitary matroids.  ...  Base packing in co-finitary matroids In this section, we prove Theorem 1.4, which is a base packing theorem for co-finitary matroids.  ...

### Page 604 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews
H is called nearly disjoint if |AN B| <1 for all distinct A,B € E(A). The Erdés-Faber-Lovasz conjecture is the following: If H is nearly disjoint on a set of size n then x'(H) <n.  ...  In this paper it is shown that if H is a nearly disjoint hypergraph on a set of size n then x'(H) < (1+ 0(1))n. The proof of this result is based on a generalization of a theorem of N.  ...

### On Packing Low-Diameter Spanning Trees

Julia Chuzhoy, Merav Parter, Zihan Tan, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming
This immediately gives a tree packing of Ω(k/log n) edge-disjoint trees of diameter at most O(k^(D(D+1)/2)).  ...  We also show that these two results are nearly tight for graphs with a small diameter: we show that there are k-edge connected graphs of diameter 2D, such that any packing of k/α trees with edge-congestion  ...  A key tool for leveraging high edge connectivity of a given graph is tree packing: a large collection of spanning trees that are (nearly) edge-disjoint.  ...

### A note on packing paths in planar graphs

András Frank, Zoltán Szigeti
1995 Mathematical programming
Analogously to Seymour's approach, we actually prove a theorem on packing cuts in an arbitrary graph and then the planar edge-disjoint paths case is obtained by planar dualization.  ...  Penn proved that the cut criterion is su cient for the existence of a near-complete packing of paths.  ...  In other words, the theorem says that there is a packing of F -good cuts which is nearly complete in the sense that each edge of K i (i = 0; : : : ; k) is covered and each but at most one edge of F j (  ...

### Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces

Tapio Rajala
2010 Monatshefte für Mathematik (Print)
We give a new estimate for the ratio of s-dimensional Hausdorff measure H s and (radius-based) packing measure P s of a set in any metric space.  ...  As an immediate consequence we improve the upper bound for the lower s-density of such sets in R n .  ...  A δ-packing of a set E ⊂ X is a countable collection of disjoint balls of radii at most δ and with centers in E.  ...

### Packing spanning graphs from separable families [article]

Asaf Ferber, Choongbum Lee, Frank Mousset
2016 arXiv   pre-print
The proof uses the local resilience of random graphs and a special multi-stage packing procedure.  ...  The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás (1976) for the case that all trees have maximum degree at most Δ.  ...  A well-known result of Böttcher, Hladký, Piguet, and Taraz  shows that in this setting, one can achieve nearly perfect packings of trees, provided that the size of the trees is bounded away from n  ...

### More-Than-Nearly-Perfect Packings and Partial Designs

David A. Grable
1999 Combinatorica
1)+o(1) uncovered t-sets, improving the earlier o(n t ) result. P v2V (H ) deg(v) = kjE(H)j=jV (  ...  edges (packings) which cover all but o(n) of the n vertices.  ...  The connection with packings in hypergraphs is that each (nearly-perfect) partial S(t; k; n) corresponds to a (nearly-perfect) packing in the following hypergraph, which we denote by H(t; k; n).  ...

### Dynamic Programming for H-minor-free Graphs [chapter]

Juanjo Rué, Ignasi Sau, Dimitrios M. Thilikos
2012 Lecture Notes in Computer Science
We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called'-triangulation.  ...  A configuration in F is a set of vertex-disjoint subgraphs F = {F 1 , . . . , F} of F .  ...  We say that a packing (that is, a collection of pairwise disjoint non-empty blocks) of the disc D k is an-packing if and only if G(⇧) does not contain K`+ 1 as a subgraph.  ...

### Packing arrays

Brett Stevens, Eric Mendelsohn
2004 Theoretical Computer Science
We introduce the consideration of a set of disjoint rows in a packing array which allows these constructions and additionally gives a new upper bound on the size of all packing arrays.  ...  We develop general direct and recursive constructions and upper bounds on the sizes of packing arrays.  ...  Tables of upper and lower bounds We include here tables showing the values for the upper and lower bounds on packing arrays.  ...

### Packing arrays

B STEVENS
2004 Theoretical Computer Science
We introduce the consideration of a set of disjoint rows in a packing array which allows these constructions and additionally gives a new upper bound on the size of all packing arrays.  ...  We develop general direct and recursive constructions and upper bounds on the sizes of packing arrays.  ...  Tables of upper and lower bounds We include here tables showing the values for the upper and lower bounds on packing arrays.  ...
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