[Facets-of-complexity] Invitation to Monday Lecture & Colloquium on June 11th 2018

[Ita Brunke <i.brunke@inf.fu-berlin.de>]

You are cordially invited to our next Monday Lecture & Colloquium on June 11th at 14:15 h & 16:00 h at FU Berlin.

Location:

Room 005 - Ground Floor
Freie Universität Berlin
Takustr. 9
14195 Berlin

Time: Monday, June 11th - 14:15 h

Lecture: John Bamberg (University of Western Australia, Perth)

Title: Bruck nets, metric planes, and their friends

Abstract:

In 1967, F. Arthur Sherk gave a simple proof that the finite metric planes (of Bachmann and Schmidt) are precisely the affine planes of odd order. Moreover, Sherk’s proof holds for a more general class of incidence structures that do not involve the ‘three-reflection theorem’ whatsoever, and thus yields a beautiful characterisation of the finite affine planes of odd order. By relaxing the first of Sherk’s axioms to ‘every pair of points lies on at most one line’, we can study what we call partial Sherk planes. In this talk, we outline our characterisation of these incidence structures as Bruck nets, in the same vein as Sherk’s result, and what it means for connected combinatorial objects such as mutually orthogonal latin squares.

(Joint work with Joanna Fawcett and Jesse Lansdown)

Coffee & Tea Break

Time: Monday, June 11th - 16:00 h

Colloquium: Anurag Bishnoi (Freie Universität Berlin)

Title: New upper bounds on some cage numbers

Abstract:

The cage problem asks for the smallest number c(k, g) of vertices in a k-regular graph of girth g. The (k, g)-graphs which have c(k, g) vertices are known as cages. While cages are known to exist for all integers k > 1 and g > 2, an explicit construction is known only for some small values of k, g and three infinite families for which g is 6, 8 or 12 and k − 1 is a prime power: corresponding to the generalized g/2-gons of order k − 1.

To improve the upper bounds on c(k, 6), c(k, 8) and c(k, 12), when k - 1 is not a prime power, one of the main techniques that has been used so far is to construct small (k, g) graphs by picking a prime power q ≥ k and then finding a small k-regular subgraph of the incidence graph of a generalized g/2-gon of order q. In this talk I will present new constructions in generalized quadrangles and hexagons which improve the known upper bound on c(k, 8) when k = p^{2h} and c(k, 12) when k = p^h, where p is an arbitrary prime. Moreover, we will see a spectral lower bound on the number of vertices in a k-regular induced subgraph of an arbitrary regular graph, which in particular will prove the optimality of a known construction in projective planes.

(Joint work with John Bamberg and Gordon Royle)

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Graduiertenkolleg 'Facets of Complexity' www.facetsofcomplexity.de/monday