arXiv subjects

Christina Zarb

Publications and source records attributed to Christina Zarb.

22 records · Page 2Linked to original sources

arXiv2014

Degree Monotone Paths

We shall study degree-monotone paths in graphs, a problem inspired by the celebrated theorem of Erd{ő}s-Szekeres concerning the longest monotone subsequence of a given sequence of numbers. A path P in a graph G is said to be a degree monotone path if the sequence of degrees of the vertices in P in the order they appear in P is monotonic. In this paper we shall consider these three problem related to this parameter: 1. Find bounds on $mp(G)$ in terms of other parameters of $G$. 2. Study $f(n,k)$ defined to be the maximum number of edges in a graph on $n$ vertices with $mp(G) < k$. 3. Estimate the minimum and the maximum over all graph $G$ on $n$ vertices of $mp(G)+mp(\overline{G})$. For the first problem our main tool will be the Gallai-Roy Theorem on directed paths and chromatic number. We shall also consider in some detail maximal planar and maximal outerplanar graphs in order to investigate the sharpness of the bounds obtained. For the second problem we establish a close link between $f(n,k)$ and the classical Turan numbers. For the third problem we establish some Nordhaus-Gaddum type of inequalities. We conclude by indicating some open problems which our results point to.

math.CO↗

arXiv2014

Independence and Matchings in $σ$-hypergraphs

Let $σ$ be a partition of the positive integer $r$. A $σ$-hypergraph $H=H(n,r,q|σ)$ is an $r$-uniform hypergraph on $nq$ vertices which are partitioned into $n$ classes $V_1, V_2, \ldots, V_n$ each containing $q$ vertices. An $r$-subset $K$ of vertices is an edge of the hypergraph if the partition of $r$ formed by the non-zero cardinalities $|K\cap V_i|, 1\leq i \leq n,$ is $σ$. In earlier works we have considered colourings of the vertices of $H$ which are constrained such that any edge has at least $α$ and at most $β$ vertices of the same colour, and we have shown that interesting results can be obtained by varying $α, β$ and the parameters of $H$ appropriately. In this paper we continue to investigate the versatility of $σ$-hypergraphs by considering two classical problems: independence and matchings. We first demonstrate an interesting link between the constrained colourings described above and the $k$-independence number of a hypergraph, that is, the largest cardinality of a subset of vertices of a hypergraph not containing $k+1$ vertices in the same edge. We also give an exact computation of the $k$-independence number of the $σ$-hypergraph $H$. We then present results on maximum, and sometimes perfect, matchings in $H$. These results often depend on divisibility relations between the parameters of $H$ and on the highest common factor of the parts of $σ$.

math.CO↗

arXiv2014

$(2,2)$-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 generalisation of Voloshin's colourings of mixed hypergraphs. We study, $χ(H)$, 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 $σ$-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 $χ(H)$ and the $(2,2)$-spectrum of $H$ simultaneously.

math.CO↗

arXiv2014

Constrained colouring and $σ$-hypergraphs

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 B{ú}jtas and Tuza, generalises both classical hypergraph colourings and the 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 $k_1 < k_2 < k_3$, $H$ would be $(α,β)$-colourable using $k_1$ and using $k_3$ colours, but not using $k_2$ colours. In an earlier paper, the first two authors introduced, for $σ$ a partition of $r$, a very versatile type of $r$-uniform hypergraph which they called $σ$-hypergraphs. They showed that, by simple manipulation of the parameters of a $σ$-hypergraph $H$, one can obtain families of hypergraphs which have $(2,r-1)$-colourings exhibiting various interesting chromatic properties. They also showed that, if the smallest part of $σ$ 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 disappearanace 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.

math.CO↗