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In honour of the work of Alex Rosa (Part I)
Org: Peter Danziger, Tommaso Traetta (Ryerson University)
[PDF]

PETER DUKES, University of Victoria
Fractional decompositions and completing partial latin squares  [PDF]

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 FRANTIĹ EK, McMaster University
d-step approach to periodical structures in strings  [PDF]

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]

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]

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]

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.