Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve

This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's book "Dense Sphere Packings" and a related preprintComment: The citations and title have been update

An Ore-type theorem for perfect packings in graphs

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4 added. To appear in the SIAM Journal on Discrete Mathematic