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Configurations and colouring problems in block designs
A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Colouring steiner quadruple systems
AbstractA Steiner quadruple system of order ν (briefly SQS(ν)) is a pair (X, B), where |X| = ν and B is a collection of 4-subsets of X, called blocks, such that each 3-subset of X is contained in a unique block of B. A SQS(ν) exists iff ν ≡ 2, 4 (mod 6) or ν = 0, 1 (the admissible integers). The chromatic number of (X, B) is the smallest m for which there is a map ϕ: X → Zm such that |ϕ(β)| ⩾ 2 for all β ϵ B. In this paper it is shown that for each m ⩾ 6 there exists νm such that for all admissible ν ⩾ νm there exists an m-chromatic SQS(ν). For m = 4, 5 the same statement is proved for admissible ν with the restriction that ν ≢ 2 (mod 12)
Switching for Small Strongly Regular Graphs
We provide an abundance of strongly regular graphs (SRGs) for certain
parameters with . For this we use Godsil-McKay
(GM) switching with a partition of type and Wang-Qiu-Hu (WQH) switching
with a partition of type . In most cases, we start with a highly
symmetric graph which belongs to a finite geometry. To our knowledge, most of
the obtained graphs are new.
For all graphs, we provide statistics about the size of the automorphism
group. We also find the recently discovered Kr\v{c}adinac partial geometry,
therefore finding a third method of constructing it.Comment: 15 page
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