Penny E. Haxell

Hall's theorem for hypergraphs

We prove a hypergraph version of Halls theorem. The proof is topological.

A condition for matchability in hypergraphs

An extension of Halls theorem is proved, which gives a condition for complete matchings in a certain class of hypergraphs.

Integer and Fractional Packings in Dense Graphs

Let be any fixed graph. For a graph G we define to be the maximum size of a set of pairwise edge-disjoint copies of in G. We say a function from the set of copies of in G to [0, 1] is a fractional -packing of G if for every edge e of G. Then is defined to be the maximum value of over all fractional -packings of G. We show that for all graphs G.

The Ramsey number for hypergraph cycles I

Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1.....,vn and edges v1v2v3, v3v4v5, v5v6v7,.....,vn-1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.

Partitioning Complete Bipartite Graphs by Monochromatic Cycles

For every positive integerrthere exists a constantCrdepending only onrsuch that for every colouring of the edges of the complete bipartite graphKn, nwithrcolours, there exists a set of at mostCrmonochromatic cycles whose vertex sets partition the vertex set ofKn, n. This answers a question raised by Erdo?s, Gyarfas, and Pyber.

The Induced Size-Ramsey Number of Cycles

For a graph H and an integer r ≥ 2, the induced r-size-Ramsey number of H is defined to be the smallest integer m for which there exists a graph G with m edges with the following property: however one colours the edges of G with r colours, there always exists a monochromatic induced subgraph H ′ of G that is isomorphic to H . This is a concept closely related to the classical r -size-Ramsey number of Erdős, Faudree, Rousseau and Schelp, and to the r -induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rodl, and Trotter. Here, we prove a result that implies that the induced r -size-Ramsey number of the cycle C l is at most c r l for some constant c r that depends only upon r . Thus we settle a conjecture of Graham and Rodl, which states that the above holds for the path P l of order l and also generalise in part a result of Bollobas, Burr and Reimer that implies that the r -size Ramsey number of the cycle C l is linear in l Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemeredi.

Tree embeddings

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k-degenerate graphs. We prove that the game chromatic index of a k-degenerate graph is at most Δ + 3k - 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges.

Bounded size components: partitions and transversals

Answering a question of Alon et al., we show that there exists an absolute constant C such that every graph G with maximum degree 5 has a vertex partition into 2 parts, such that the subgraph induced by each part has no component of size greater than C. We obtain similar results for partitioning graphs of given maximum degree into k parts (k >2) as well. A related theorem is also proved about transversals inducing only small components in graphs of a given maximum degree.

On the Strong Chromatic Number

Let

A Note on Vertex List Colouring

G