Speaker: Latifa Zekaoui
Title: Mixed Covering Arrays on Graphs
Abstract:

Covering arrays are generalizations of orthogonal arrays that are used for
testing software, networks and circuits.  Let N, g, k be positive integers. 
A covering array is a k by N array with entries from Z_g such that any 2 by 
N subarray contains every ordered pair of elements from Z_g in some of its columns.

In this talk, we look at a further generalization of covering arrays that 
considers both mixed alphabet sizes for different rows and a graph structure 
on the rows that prescribes the pair of rows for which we require the covering property.
A (standard) covering array is a particular case where all alphabets are the same and G 
is the complete graph. We give optimal constructions of mixed covering arrays for trees, 
cycles and the bipartite graphs.

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Speaker: Paul Elliott Magwood
Title: Constructing optimal solutions to the minimum cost 2-edge-connected 
spanning subgraph problem with metric costs
Abstract:

Given a complete graph on n vertices and nonnegative costs assigned
to the edges, the 2-edge-connected Spanning Subgraph Problem (2EC) is 
that of finding a minimum cost 2-edge-connected graph which spans all 
n vertices. Monma, Munson, and Pulleyblank showed that if the costs 
assigned to the edges of the complete graph are metric then there 
exists an optimal solution to the 2EC which is edge-minimally 
2-edge-connected, is 2-vertex-connected, has maximum valency 3, and 
the removal of any pair of edges leaves a bridge in at least one of 
the remaining components. For this presentation, we will show how to 
construct all the graphs which have these properties using their ear 
decompositions, their necklaces, and their min-cut cactii.

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Speaker: Shonda Gosselin
Title: Vertex-Transitive Self-Complementary 3-Hypergraphs
Abstract:

A 3-hypergraph is a pair (V,E) in which V is a finite set or vertices and E is a set
of 3-subsets of V called edges. An isomorphism between the 3-hypergraphs X=(V,E) and
Y=(W,F) is a bijection from V to W which induces a bijection between E and F.  If
such a bijection exists, we say that X and Y are isomorphic.  An automorphism of X
is an isomorphism from X to X.
 The complement X^C of  a 3-hypergraph X=(V,E) is the 3-hypergraph
X^C=(V,E^C), in which E^C is the set of all 3-subsets of V which are not in E.  A
3-hypergraph X is called self-complementary if it is isomorphic to its complement
X^C.  A 3-hypergraph X=(V,E) is vertex-transitive if for any two vertices v,w in V,
there exists an automorphism of X which maps v to w.
     
In this talk, some results regarding the possible orders of
vertex-transitive self-complementary 3-hypergraphs will be discussed, and some
constructions will be presented. In particular, we will investigate the structure of
vertex-transitive self-complementary 3-hypergraphs of prime order.

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Speaker: Gabriel Verret
Title : Shifts in Cayley Graphs
Abstract : 

An automorphism of a simple graph is called a shift if it maps every vertex to an
adjacent one. We consider which Cayley graphs have shifts and also for which groups
all Cayley graphs on these groups admit a shift.

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Speaker: Andrea Burgess
Title: Colouring even cycle systems
Abstract:

An m-cycle system of order n is a decomposition of the complete graph K_n into
m-cycles.  An m-cycle system is said to be weakly
k-colourable if its vertices can be partitioned into k colour classes such that no
cycle has all of its vertices the same colour.  A cycle system's chromatic number is
the smallest value of k for which the system is weakly k-colourable.  While
colourings of 3-cycle systems, or Steiner triple systems, have been widely studied,
less is known regarding colourings of m-cycle systems in general.  In this talk, we
present some results on weak colourings of m-cycle systems for which the cycle
length m is even, in particular, the result that for any integers r and k, both
greater than or equal to 2, , there is a k-chromatic (2r)-cycle system.  We
illustrate with examples of constructions of cycle systems with prescribed chromatic
number.  This is joint work with David Pike, Memorial University of Newfoundland.