Christina Zarb

Constrained Colouring and σ-Hypergraphs

Abstract A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bujtás and Tuza, generalises both classical hypergraph colourings and more general Voloshin colourings of hypergraphs. In fact, for r-uniform hypergraphs, classical colourings correspond to (2, r)-colourings while an important instance of Voloshin colourings of r-uniform hypergraphs gives (2, r −1)-colourings. One intriguing aspect of all these colourings, not present in classical colourings, is that H can have gaps in its (α, β)-spectrum, that is, for k1 < k2 < k3, H would be (α, β)-colourable using k1 and using k3 colours, but not using k2 colours. In an earlier paper, the first two authors introduced, for being a partition of r, a very versatile type of r-uniform hypergraph which they called -hypergraphs. They showed that, by simple manipulation of the param- eters of a σ -hypergraph H, one can obtain families of hypergraphs which have (2, r − 1)-colourings exhibiting various interesting chromatic proper- ties. They also showed that, if the smallest part of is at least 2, then H will never have a gap in its (2, r − 1)-spectrum but, quite surprisingly, they found examples where gaps re-appear when α = β = 2. In this paper we extend many of the results of the first two authors to more general (α, β)-colourings, and we study the phenomenon of the disappearance and re-appearance of gaps and show that it is not just the behaviour of a particular example but we place it within the context of a more general study of constrained colourings of σ -hypergraphs.

The Saturation Number for the Length of Degree Monotone Paths

Abstract A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) < k, while mp(G+e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.

( 2 , 2 ) -colourings and clique-free σ -hypergraphs

We consider vertex colourings of r -uniform hypergraphs H in the classical sense, that is such that no edge has all its vertices given the same colour, and ( 2 , 2 ) -colourings of H in which the vertices in any edge are given exactly two colours. This is a special case of constrained colourings introduced by Bujtas and Tuza which, in turn, is a generalization of Voloshins colourings of mixed hypergraphs. We study, ? ( H ) , the classical chromatic number, and the ( 2 , 2 ) -spectrum of H , that is, the set of integers k for which H has a ( 2 , 2 ) -colouring using exactly k colours.We present extensions of hypergraphs which preserve both the chromatic number and the ( 2 , 2 ) -spectrum and which, however often repeated, do not increase the clique number of H by more than a fixed number. In particular, we present sparse ( 2 , 2 ) -colourable clique-free ? -hypergraphs having arbitrarily large chromatic number-these r -uniform hypergraphs were studied by the authors in earlier papers. We use these ideas to extend some known 3 -uniform hypergraphs which exhibit a ( 2 , 2 ) -spectrum with remarkable gaps. We believe that this work is the first to present an extension of hypergraphs which preserves both ? ( H ) and the ( 2 , 2 ) -spectrum of H simultaneously.

Ramsey numbers for degree monotone paths

Abstract A path v 1 , v 2 , … , v m in a graph G is degree-monotone if d e g ( v 1 ) ≤ d e g ( v 2 ) ≤ ⋯ ≤ d e g ( v m ) where d e g ( v i ) is the degree of v i in G . Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by M k ( m ) the minimum number M such that for all n ≥ M , in any k -edge coloring of K n there is some 1 ≤ j ≤ k such that the graph formed by the edges colored j has a degree-monotone path of order m . We prove several nontrivial upper and lower bounds for M k ( m ) .

Selective hypergraph colourings

We look at colourings of r -uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge E in an r -uniform hypergraph H whose vertices are coloured is the partition of r induced by the colour classes of the vertices in E . Let Q be a set of partitions of r . A Q -colouring of H is a colouring of its vertices such that only patterns appearing in Q are allowed. We first show that many known hypergraph colouring problems, including Ramsey theory, can be stated in the language of Q -colourings. Then, using as our main tools the notions of Q -colourings and Σ -hypergraphs, we define and prove a result on tight colourings, which is a strengthening of the notion of unique colourability. Σ -hypergraphs are a natural generalisation of ? -hypergraphs introduced by the first two authors in an earlier paper. We also show that there exist Σ -hypergraphs with arbitrarily large Q -chromatic number and chromatic number but with bounded clique number. Dvořak et?al. have characterised those Q which can lead to a hypergraph with a gap in its Q -spectrum. We give a short direct proof of the necessity of their condition on Q . We also prove a partial converse for the special case of Σ -hypergraphs. Finally, we show that, for at least one family Q which is known to yield hypergraphs with gaps, there exist no Σ -hypergraphs with gaps in their Q -spectrum.