Deryk Osthus

The minimum degree threshold for perfect graph packings

AbstractLet H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H,n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G)≥k contains a perfect H-packing. We show that

Popularity based random graph models leading to a scale-free degree sequence

\delta (H,n) = \left( {1 - \frac{1} {{\chi ^ * (H)}}} \right)n + O(1)

On random planar graphs, the number of planar graphs and their triangulations

.The value of χ*(H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies χ(H)−1

Edge-disjoint Hamilton cycles in random 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 Diracs theorem on Hamilton cycles and Tuttes 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 Diracs 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.

Loose Hamilton cycles in hypergraphs

As an extremely simplified model for the growth of the world wide web, Dorogovtsev et al. (Phys. Rev. Lett. 85 (2000) 4633) and Drinea et al. (Variations on random graph models for the web, Technical Report, Department of Computer Science, Harvard University, 2001) introduced the following random graph model, which generalizes an earlier model of Barabasi and Albert (Science 286 (1999) 509): at each time step we add a new vertex incident to r edges. The other endpoints of these edges are chosen with probability proportional to their in-degrees plus an initial attractiveness ar, where a is a constant. For all a, r@?N, we determine the asymptotic form of the degree distribution for most of the vertices. Confirming non-rigorous arguments of Dorogovtsev et al. and Drinea et al., this shows that for such a, the proportion P(d) of vertices of degree d almost surely obeys a power law, where P(d) is of the form d^-^2^-^a for large d. The case a=1 (which corresponds to the model of Barabasi and Albert) was proved earlier by Bollobas et al. (Random Struct. Algorithms 18 (2001) 279).

An exact minimum degree condition for Hamilton cycles in oriented graphs

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than (n-12)-(2n/32), then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d=

Hamilton ℓ-cycles in uniform hypergraphs

Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |Pn| ≤ n! (75.8)n+o(n) and Bender, Gao and Wormald proved that |Pn| ≥ n! (26.1)n+o(n). Gerke and McDiarmid proved that almost all graphs in Pn have at least 13/7n edges. In this paper, we show that |Pn| ≤ n! (37.3)n+o(n) and that almost all graphs in Pn have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least e3(3n-m)/2 labelled triangulations on n vertices, where e is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.

Embeddings and Ramsey numbers of sparse κ -uniform hypergraphs

We show that provided log50n/ni¾?pi¾?1-n-1/4log9n we can with high probability find a collection of i¾?i¾?G/2i¾? edge-disjoint Hamilton cycles in G~Gn,p, plus an additional edge-disjoint matching of size i¾?n/2i¾? if i¾?G is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich.