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Org: Peter Danziger, Tommaso Traetta (Ryerson University)
[PDF]
- PETER DUKES, University of Victoria
Fractional decompositions and completing partial latin squares [PDF]
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A fractional triangle decomposition of a graph $G$ is a nonnegative weighting of the copies of $K_3$ in $G$ such that every edge has total weight $1$.
I will survey some sufficient conditions for a graph $G$ to admit a fractional triangle decomposition. The focus is on graphs which are `locally dense' in the complete graph or complete tripartite graph. Using a recent theorem of Barber et al., this leads to asymptotic results on genuine triangle decompositions.
An application is the completion problem for partial latin squares, a topic investigated in Alex Rosa's work.
- FRANÄšK FRANTIÅ EK, McMaster University
d-step approach to periodical structures in strings [PDF]
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What is the maximum number of runs/distinct squares in a string of length $n$ are natural questions. The runs conjecture limiting the number of runs to $n$ was settled in 2015, the equivalent squares conjecture is not settled yet; the best upper bound is 11n/6. In 2013, we proposed d-step approach to both problems and conjectured tighter upper bounds for both. Using the technique of Lyndon roots, we showed recently that d-step conjecture for runs holds true. The d-step approach has been used for computing the values of respective functions for strings of lengths that were previously unfeasible.
- ESTHER LAMKEN, University of Caltech
An existence theory for incomplete designs [PDF]
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An incomplete pairwise balanced design is equivalent to a pairwise balanced
design with a distinguished block, called a `hole'. If there are $v$
points, a hole of size $w$, and all (other) block sizes equal $k$, this is
denoted IPBD$((v;w),k)$. In addition to congruence restrictions on $v$ and
$w$, there is also a necessary inequality: $v >(k-1)w$. In this talk, I will
discuss existence results for IPBD$((v;w),k)$: one in which $w$
is fixed and $v$ is large, and the other in the case $v > (k-1+\epsilon) w$
when $w$ is large (depending on $\epsilon$). I will also discuss some
generalizations.
- NABIL SHALABY, Memorial University
Rosa sequences [PDF]
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In 1966 Rosa introduced a sequence R = $(r_1, r_2, . . . , r_{2n+1} )$ of 2n+1 integers that satisfy the following conditions:
(1) For every $k \in$ $\{1,2, ...,n \}$ there exist exactly two elements $r_i ,r_j$ such that $r_i= r_j = k$. \
(2) If $r_i= r_j= k$ with $i < j$, then $j - i = k$. \
(3) $r_{n+1} = 0 $ \
For example, (1,1,3,4,0,3,2,4,2) is a Rosa sequence of order 4 . \
In this talk, we survey the development of Rosa sequences and reflect on examples of Professor Rosa as a supervisor.
- DOUG STINSON, University of Waterloo
Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets [PDF]
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A $(t, s, v)$-all-or-nothing transform (AONT) is a bijective mapping defined
on $s$-tuples over an alphabet of size $v$, which satisfies the condition
that the values of any $t$ input co-ordinates are
completely undetermined, given the values of any $s-t$ output co-ordinates.
The main question we address in this talk is:
for which choices of parameters does a $(t, s, v)$-AONT exist?
We concentrate on the case $t=2$ for arbitrary values of $v$, where we
obtain various necessary as well as sufficient conditions for existence of these objects.
We also show some connections between AONTs, orthogonal arrays and resilient functions.
