Prairie Discrete Math Workshop
University of Calgary
May 4 & 5, 2012
Conference Program
Conference Sponsors
The Pacific Institute for the Mathematical Sciences
Department of Mathematics and Statistics (University of Calgary)
Faculty of Science (University of Calgary)
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Welcome to the 2012 Prairie Discrete Math Workshop
It is our great pleasure to welcome you to the 2012 Prairie Discrete Math
Workshop (PDMW). PDMW provides an excellent opportunity for networking and joint research with collaborators in the prairie region (Manitoba,
Saskatchewan and Alberta), as well as neighbouring provinces and states.
Previously, it has been held at the Universities of Regina (2002, 2011), Lethbridge (2003, 2006), Manitoba (2008, 2010), Winnipeg (2005), and at UBCOkanagan (2009). This years workshop is hosted by the University of Calgary
in Calgary, Alberta, and will cover a broad variety of topics of great importance in discrete mathematics. Topics include graph theory, combinatorics,
design theory, combinatorial game theory, discrete geometry and algorithms.
The program includes six invited speakers and four contributed talks spanning a period of two days.
To all participants, we wish you a very enjoyable workshop and pleasant
stay in Calgary!
Karen Seyffarth
& Michael Cavers
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Invited Speakers
• Gary MacGillivray
University of Victoria
Cops and Robber: Who Wins and How Fast?
• Karen Meagher
University of Regina
Erdős-Ko-Rado Theorem for Permutations
• Ortrud Oellermann
University of Winnipeg
Convexity in Graphs
• Paul Ottaway
Thompson Rivers University
Winning and Losing Combinatorial Games
• Lorna Stewart
University of Alberta
Overlap Numbers of Graphs
• Csaba Tóth
University of Calgary
Packing Anchored Rectangles
Conference Organizers
• Karen Seyffarth, Associate Professor, University of Calgary
• Michael Cavers, NSERC Postdoctoral Fellow, University of Calgary
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List of Speakers and Abstracts
Graham Banero, Simon Fraser University
Abelian and Additive Complexity:
Analyzing Structure in Infinite Words
Motivated by the problem of avoidability of abelian and additive squares,
notions of the abelian and additive complexity of an infinite word w over
a finite alphabet have been introduced as follows. The abelian complexity
ρab (w, n) counts the maximum number of factors of w having length n, such
that no two are permutations of each other; the additive complexity ρΣ (w, n)
counts the number of distinct sums of factors of w with length n. We give an
overview of some main results on these relatively new concepts, paying attention in particular to the rich similarities and surprising differences between
the two.
Steven Chaplick, Wilfrid Laurier University
Edge Intersection Graphs of L-Shaped Paths in Grids
In this work we continue the study of the edge intersection graphs of single
bend paths on a rectangular grid (i.e., the edge intersection graphs where each
vertex is represented by one of the following shapes: x, p, y, q). These graphs,
called B1 -EPG graphs, were first introduced by Golumbic et al (2009).
We focus on the class [x] (the edge intersection graphs of x-shapes) and
show that testing for membership in [x] is NP-Complete. A characterization
and polytime recognition algorithm for special subclasses of Split ∩ [x] is then
given.
We also consider the natural subclasses of B1 -EPG formed by the subsets
of the four single bend shapes (i.e., {x}, {x, p}, {x, q}, {x, p, q} – note: all other
subsets are isomorphic to these up to 90 deg rotation). Some properties of
these classes as well as the expected strict inclusions and incomparability
(i.e., [x] ( [x, p], [x, q] ( [x, p, q] ( B1 -EPG and [x, p] is incomparable with
[x, q]) are presented.
Joint work with Kathie Cameron and Chı́nh T. Hoàng. Research support
by Natural Sciences and Engineering Research Council of Canada.
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Gary MacGillivray, University of Victoria
Cops and Robber: Who Wins and How Fast?
Cops and Robber is a discrete-time pursuit game played on a graph: a number
of cops chase a robber from vertex to vertex with the goal of catching him in
the sense that some cop eventually occupies the same vertex as the robber.
We will survey results concerning whether the cops can catch the robber and,
if so, how long that will take assuming both sides play optimally.
Karen Meagher, University of Regina
Erdős-Ko-Rado Theorem for Permutations
Two permutations in the symmetric group on n vertices can be considered
to be “intersecting” if they both map some i ∈ {1, ..., n} to the same element
(we say the permutations agree on i). With this definition, what is the largest
set of permutations such that any two are intersecting? In the last 10 years,
several different proofs have been published that show that the largest such
set is either the stabilizer of a point or a coset of the stabilizer of a point.
This result is known as the EKR theorem for permutations.
There are several questions that are natural to ask once this result is established. For example what is the largest set of permutations that agree on
a set of t elements? There are other ways to define intersection for permutations, does an EKR type result hold for these as well? What is the largest
set of intersecting permutations in a subgroup of the symmetric group? I will
present some new results by various researchers on these questions.
Shahla Nasserasr, University of Regina
TPk Completion of Partial Matrices with One Unspecified Entry
An m × n matrix is called TPk if every minor of size at most k is positive.
The TPk completion problem for patterns of specified entries is considered.
For a given pattern with one unspecified entry, the minimal set of conditions
characterizing TPk completability is given. These conditions are finitely many
polynomial inequalities in the specified entries of the pattern. This is joint
work with C. Johnson and V. Akin.
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Ortrud Oellermann, University of Winnipeg
Convexity in Graphs
Convexity in Euclidean space is used to motivate the definition of abstract
convexities. We describe how the idea of an interval naturally leads to several
graph convexities in graphs. In this talk we focus on graph classes for which
a specified graph convexity possesses a property related to convexity such as,
for example, the Krein-Milman property and the separation properties. As
part of our discussions we include several open problems.
Paul Ottaway, Thompson Rivers University
Winning and Losing Combinatorial Games
In this talk, we shall give an overview of the field of combinatorial game
theory. In particular, we will examine the similarities and differences between
the normal play and misere play conventions where the last player to make a
move wins or loses, respectively. Finally, the concept of misère quotients will
be introduced as an emerging technique for analysis.
Andrew Poelstra, Simon Fraser University
On the Existence of Double 3-Term Arithmetic Progressions
Van der Waerden’s Theorem states that for any number of colors r, and any
length k, every r-coloring of the natural numbers must contain a monochromatic k-term arithmetic progression (a sequence x1 , x2 , . . . , xk with constant
gap between consecutive entries).
We extend this idea to double k-term arithmetic progressions: sets
xi1 , xi2 , . . . , xik
which have constant gap between consecutive entries, and constant gap between consecutive indices.
We ask whether every r-coloring of the natural numbers must contain a kterm arithmetic progression, present computational evidence to suggest that
the answer is (in general) no, and show that this question is equivalent to
one about infinite words on finite alphabets.
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Lorna Stewart, University of Alberta
Overlap Numbers of Graphs
An overlap representation of a graph is an assignment of sets to the vertices
of the graph in such a way that two vertices are adjacent if and only if the
sets assigned to them intersect and neither set is contained in the other. The
overlap number of a graph is the minimum number of elements needed to
form such a representation. We discuss overlap representations and overlap
numbers in general and in cases where the graphs or the sets are required to
have certain properties.
Csaba Tóth, University of Calgary
Packing Anchored Rectangles
Let S be a set of n points in the unit square [0, 1]2 , one of which is the origin.
We construct n pairwise interior-disjoint axis-aligned empty rectangles such
that the lower left corner of each rectangle is a point in S, and the rectangles
jointly cover at least a positive constant area (about 0.09). It is a longstanding
conjecture that the rectangles in such a packing can always cover a total area
of at least 1/2, but no positive constant lower bound has been known.
(Joint work with Adrian Dumitrescu, University of Wisconsin—Milwaukee.)
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Conference Schedule
Time
Friday, May 4, 2012 Saturday, May 5, 2012
8:30 - 9:30
Breakfast
Breakfast
9:30 - 10:30
Csaba Tóth
Gary MacGillivray
10:30 - 11:00
Coffee Break
Coffee Break
11:00 - 12:00
Lorna Stewart
Paul Ottaway
12:00 - 14:00 Lunch (provided)
Lunch (provided)
14:00 - 14:30
Steven Chaplick
Graham Banero
14:30 - 15:00 Andrew Poelstra
Shahla Nasserasr
15:00 - 15:30
Coffee Break
Coffee Break
15:30 - 16:30
Karen Meagher
Ortrud Oellermann
Registration and breaks for the workshop will take place in the
Department of Mathematics & Statistics Lounge
located on the 4th floor (MS 461) of the Math Sciences building.
Talks will take place in room MS 431.
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