


Graph Theory
Org: Jing Huang, Kieka Mynhardt and Wendy Myrvold (Victoria) [PDF]
 RICHARD ANSTEE, UBC Mathematics, #1211984 Mathematics Rd, Vancouver, BC
Robustness of the property of being matchable subject to
vertex deletion
[PDF] 
We consider classes of graphs which are easily seen to have many
perfect matchings. The class of grid graphs is our main example. We
then consider what properties to impose on choosing a subset of
vertices A Í V(G) for vertex deletion in a graph G (from
such a class) so that the vertex deleted subgraph GA has a perfect
matching. Certain conditions are easy. An even number of vertices
must be deleted. If the graph is bipartite then the deleted vertices
must have equal numbers from both parts of the bipartition. Also one
cannot delete all the neighbours of a given vertex.
We obtain two results. In one, the deleted vertices are confined to
the periphery of the graph and in the other the deleted vertices are
required to be far apart. The motivation was a result of Jamison and
Lockner presented at CGTC 34 (2003) but we can see this as an
alternative to the edge extendibility investigations of Plummer
et al.
 ANTHONY BONATO, Wilfrid Laurier University, Waterloo, ON N2L 3C5
Random graph models and the web
[PDF] 
Much recent attention has focussed on the rigorous design and analysis
of models for the web graph and other massive selforganizing
networks. Models for these networks often incorporate copying in
their design, where a new node imperfectly duplicates some of the link
structure of an existing node. The motivation for copying models
comes from the fact that each node acts as an independent agent, which
will base its decision on how to link to the existing network on local
knowledge. As a result, the neighbourhood of a new node will often be
similar to that of an existing node. Consideration of copying models
and their limiting behaviour have led to the discovery of new
connections between random graphs, graph homomorphisms, vertex pursuit
games, and infinite graphs.
This is joint work with Manuel Bodirsky, Peter Cameron, Dejan
Deli\'c, Jeannette Janssen, and Changping Wang.
 RICK BREWSTER, Thompson Rivers University, Box 3010, Kamloops, BC V2C 3N5
Switch invariant homomorphism targets
[PDF] 
A redblue graph G is a set of vertices V(G) together with two
edges sets: E_{r}(G), the red edges; and E_{b}(G), the blue edges.
Given redblue graphs G and H, a homomorphism of G to H
is a function f : V(G) ® V(H) such that uv Î E_{i}(G)
implies f(u)f(v) Î E_{i}(H), for i Î {r,b}. We write G ®H.
Given a redblue graph G and a vertex v Î V(G), the graph G^{v}
is obtained from G by switching the colour of edge incident
with v. (This process is analogous to Seidel switching when G is
a complete redblue graph.) A redblue graph H is switch
invariant if for any redblue graph G we have G ® H if and
only if G^{v} ® H for all v Î V(G). Switch invariant graphs
arise in modeling certain constraint satisfaction problems.
We present a characterization of switch invariant redblue graphs,
plus generalize the problem to edgecoloured graphs with m edge
colours.
This is joint work with Timothy Graves.
 SEAN DAUGHERTY, University of Victoria
Exploring the Influence and Total Influence Numbers of a
Graph
[PDF] 
Recently, the study of social networks has given rise to a graph
parameter known as the influence number: h(G). This
parameter measures the influence that a vertex subset S has on the
remaining vertices such that each vertex not in S is influenced by
the closest member of S as follows: h(S) = å_{u Ï S}2^{d(u,S)}. The influence number of a graph is the maximum such
value over all possible vertex subsets: h(G) = max_{S Í V}h(S). A natural extension to the influence number allows each
vertex not in S to be influenced by every vertex in S. This gives
the total influence number: h_{T}(G) = max_{S Í V}[å_{u Î S} å_{v Ï S} 2^{d(u,v)}]. Other applications
include facility location problems where the quality of service decays
exponentially with respect to distance. This talk will explore some
of the basic results involving the influence and total influence
numbers of a graph.
 DANNY DYER, Memorial University of Newfoundland, St. John's, NL, A1C 5S7
Time constrained searching and sweeping
[PDF] 
Graph searching and graph sweeping are often used as real world models
in which mobile agents move to capture a mobile intruder. In graph
searching, the intruder can only be located on vertices, while in
graph sweeping, the intruder can also be located on edges. The most
common problem examined with these models is to find the fewest agents
which can guarantee the intruder's capture. However, in some
situations, agents are "cheap", and we may then insist the intruder
be captured quickly. We examine several such cases in variations of
the the graph searching and sweeping models.
 STEPHEN FINBOW, St. Francis Xavier University, PO Box 5000, Antigonish, Nova
Scotia B2G 2W5
Generalisations of Domination, Independence and Irredundance
[PDF] 
The study of the concepts of domination, independence and irredundance
has led to many generalisations of these concepts in the literature.
We consider a method for categorising many of these generalisations
into a general framework. The framework is used to find connections
between the generalised concepts and allows for many theorems
(regarding these connections) to be proven at once.
 SHANNON FITZPATRICK, University of Prince Edward Island
Independent Domination, Domination and Chromatic Number
[PDF] 
An independent dominating set D is any set of vertices in a graph
G such that no vertices in D are adjacent, but every vertex in
V(G)  D is adjacent to a vertex in D. The independent domination
number of G, denoted i(G), is the cardinality of a minimum
independent dominating set.
The following upper bounds have been found by Bollobás & Cockayne
and MacGillivray & Seyffarth, respectively:

i(G) £ n  g(G)  
é ê


ng(G)
g(G)


ù ú

+ 1 
 
 i(G) £ n  c(G)  
é ê


nc(G)
c(G)


ù ú

+ 1 
 
where n=V(G), g(G) is the domination number of G and
c(G) is the chromatic number of G. I will discuss the
characterization of those graphs for which equality is achieved for
the first upper bound and the problem of determining those instances
for which the first upper bound provides a better result than the
second.
 PATRICK FOWLER, University of Sheffield, UK
Cospectrality and the fullerenes
[PDF] 
A fullerene polyhedron, representing an allcarbon molecule
C_{n}, is cubic, with only pentagonal and hexagonal faces
(12 and n/210, respectively, by Euler's theorem). The spectrum
of the adjacency matrix contains a great deal of qualitative chemical
information on optimum electron count, electronic configuration,
stability and reactivity of the molecule. Connections between graph
theory and these chemical models will be discussed. Cospectral
fullerene graphs would be of interest as representing molecules with
structures that were different but with (some) identical properties.
As yet, no cospectral fullerenes are known, but many pairs of
fullerenes with cospectral duals have now been found
(P. W. Fowler and M. Duncan, to be published). The dual of a
fullerene graph is also a polyhedral graph, having all faces
triangular, 12 vertices of degree 5 and all others of degree 6;
considered as the skeleton of a molecule, it is a model for large
boron hydrides, based on extrapolation of the known clososeries of
borane anions, B_{N} H_{N}^{2}. A construction for
(infinite) families of cospectral fullerene duals will be presented.
 LUIS GODDYN, Simon Fraser University, Burnaby, BC V5A 1S6
Silver Cubes
[PDF] 
An n ×n matrix is silver if, for i=1,...,n, every
symbol in {1,2,...,2n1} appears as an entry in either row i
or column i.
An IMO 1997 question introduced this definition, and asked whether a
silver matrix of order 1997 exists. (In fact, one exists if and only
if n=1 or n is even.) In this paper we investigate higher
dimensional analogs, silver cubes.
Finding the correct generalization is the first challenge. The cells
on the main diagonal of a silver matrix are treated specially. What
should serve as a "diagonal" in a ddimensional n×n×¼×n silver cube? We propose that a "diagonal" should
be a "maximum independent set in the dth cartesian power of the
complete graph of order n." This definition is motivated by
"minimal defining sets" for colouring such graphs. The challenge is
to label the cells with symbols 1,2,...,d(n1)+1 so that, for each
cell c on the "diagonal", every symbol appears once in one of the
d(n1)+1 cells orthogonally translated from c. We present
constructions, nonexistence proofs and connections with coding theory
and projective geometry.
This is joint work with M. Ghebleh, E. Mahmoodian and
M. VerdianRizi.
 GENA HAHN, Université de Montréal
Recent bounds on the injective chromatic number
[PDF] 
We introduced the injective chromatic number in [1] but it had
been studied in different guises before. In this talk we review
previous results and give new bounds on the number for some classes of
graphs.
This is joint work with Alain Doyon, André Raspaud and Weifan Wang.
References
 [1]

G. Hahn, J. Kratochvíl, D. Sotteau and J. Sirán,
On injective colourings.
Discrete Math. 256(2002), 179192.
 BERT HARTNELL, Saint Mary's University, Halifax, Nova Scotia B3H 3C3
Monitoring a Network with fewer Watchpeople: the trade offs
[PDF] 
A dominating set S of a graph is a set of vertices such that every
vertex of the graph is either in S or adjacent to some vertex
of S. In particular, the dominating set is considered static and
all vertices of the graph are dominated all of the time. In many
applications this may not be practical as resources are simply too
limited. Hence we might allow vertices to not be dominated for
certain periods of time as long as such time intervals are not
excessive. Thus we could allow each member of our set to either
remain fixed or move to an adjacent vertex in one unit of time. What
can be said about the time vertices are not monitored if the
dominating set is cut to some fraction of the original number required
or by a fixed number? Although more questions than answers will be
given, some observations will be outlined.
 NORA HARTSFIELD, Western Washington University, Bellingham, WA 982259063,
USA
Selfdual embeddings of strongly regular selfcomplementary
graphs
[PDF] 
An infinite family of selfdual embeddings of strongly regular
selfcomplementary graphs is presented. These embeddings are very
highly structured and have the property that if one draws the dual on
the same surface, without moving any vertices, the resulting embedding
of the complement is also selfdual and all four embeddings are map
isomorphic.
 PETER HORAK, University of Washington, Tacoma
Perfect dominating sets
[PDF] 
A perfect dominating set S of distance d in a graph G is a set
of its vertices so that each vertex of G is at distance at most d
from exactly one vertex of S. In 1968 Golomb and Welsh conjectured
(they formulated their conjecture in terms of Leedistance error
correcting codes) that a perfect dominating set of distance d in the
ndimensional torus C_{k} ×¼×C_{k} exists only in
the case for n=1 and any d or n=2 and any d or d=1 and any
n. Despite a lot of effort by researchers both in graph theory and
in coding theory the conjecture is still open although there are
plenty of partial results supporting the conjecture. In this talk we
will discuss some variations of the conjecture that are motivated by
an application in computer science.
 ROSS KANG, Oxford University, Department of Statistics, 1 South Parks
Road, Oxford OX1 3TG, United Kingdom
The timproper chromatic number of random graphs
[PDF] 
We consider the tdependence and timproper chromatic numbers of
the ErdösRényi random graph. As usual, G_{n,p} denotes a
random graph with vertex set [n] = {1,...,n} in which the edges
are chosen independently at random with probability p. The
tdependence number a^{t}(G) of a graph G is the maximum size
of a tdependent seta vertex subset which induces a subgraph of
maximum degree at most t. The timproper chromatic number
c^{t}(G) is the smallest number of colours needed in a timproper
colouringa colouring of the vertices in which colour classes are
tdependent sets. Note that c^{t}(G) ³ V(G)/a^{t}(G) and
c^{t}(G) ³ c^{t+1}(G) for any graph G.
Clearly, when t = 0, we are considering the independence number and
chromatic number of random graphs, and the problem of determining the
asymptotic behaviour of the chromatic number c(G_{n,p}) was once
one of the central open problems in random graph theory. We consider
a fixed edgeprobability p where 0 < p < 1, and t=t(n). Our
results on c^{t}(G_{n,p}) break into three ranges for t(n). We
say that a property holds asymptotically almost surely (a.a.s.) if it
holds with probability ® 1 as n ® ¥.
(a) If t(n) = o(logn), then c^{t}(G_{n,p}) ~ ([ 1/2]log[ 1/(1p)]) [(n)/(logn)] a.a.s. Thus, if
the impropriety t does not grow too fast, then the timproper
chromatic number is likely to be close to the proper chromatic
number.
(b) If t(n) = Q(logn), then c^{t}(G_{n,p}) = Q([(np)/(t)]) a.a.s.
(c) Lastly, if t(n)/logn ® ¥, then
c^{t}(G_{n,p}) ~ [(np)/(t)] a.a.s.
This is joint work with Colin McDiarmid.
 BILL KOCAY, University of Manitoba
Degree Sequence Problems for 3Hypergraphs
[PDF] 
The ErdösGallai conditions are necessary and sufficient
conditions for the existence of a simple graph with a given degree
sequence. Much work has been done characterizing the polytope of
degree sequences of simple graphs. The corresponding conditions for
3hypergraphs are still unknown.
A simple 3hypergraph G consists of a set V of vertices and E of
edges, such that each edge is a triple u,v,w of distinct vertices.
Repeated triples are not allowed in G. The degree of a vertex v
is deg(v), the number of triples containing v. The degree
sequence of G is the sequence of degrees D(G) = (d_{1},d_{2},...,d_{n}),
such that d_{1} ³ d_{2} ³ ¼ ³ d_{n}. We ask when a given
sequence D is the degree sequence of a simple 3hypergraph?
It is still unknown whether this problem has a
polynomialtime algorithmic solution, or whether it is NPcomplete.
Recently Kocay and Li showed that any two 3hypergraphs
with the same degree sequence can be transformed into
each other by a sequence of operations known as trades.
The proof is based on nullhypergraphs. We describe the
structure of nullhypergraphs, and a closely related NPcomplete
problem for 3hypergraph degree sequences.
 JIM LIU, University of Lethbridge
On the recognition of probe graphs of some selfcomplementary
classes of perfect graphs
[PDF] 
In this paper we consider the recognition of some probe graph classes.
Given a class of graphs G, a graph G is a probe
graph of G if its vertices can be partitioned into a
set P of probes and an independent set N
of nonprobes, such that G can be extended to a graph of
G by adding edges between certain nonprobes. We show that
there are polynomialtime recognition algorithms for probe cographs,
probe P_{4}reducible graphs, probe P_{4}sparse graphs, and probe
splitgraphs.
Joint work with MawShang Chang, Ton Kloks, Dieter Kratsch and
ShengLung Peng.
 JOY MORRIS, University of Lethbridge, Lethbridge, AB T1K 6R4
Cyclic Hamiltonian (Near)Decompositions of the Complete Graph
[PDF] 
It has been proven that the complete graph on n vertices can be
decomposed into Hamiltonian cycles, whenever n is odd. Similarly,
if we remove a 1factor (perfect matching) from the complete graph on
an even number of vertices, the remaining graph can always be
decomposed into Hamiltonian cycles; this is what is referred to as a
neardecomposition. To make the problem interesting again, we put
constraints on the Hamiltonian cycles that we allow to be in the
decomposition. A cyclic Hamiltonian decomposition of the complete
graph is a decomposition of the complete graph into Hamiltonian
cycles, in such a way as to ensure that rotating any cycle in the
decomposition gives us a (possibly different) cycle in the
decomposition. It has been proven that a cyclic Hamiltonian
decomposition of the complete graph on n vertices always exists when
n is odd, as long as n ¹ 15 and n ¹ p^{a}, where p
is prime and a > 1, and that these constraints are necessary.
We prove that when n is even, cyclic Hamiltonian neardecompositions
of the complete graph on n vertices exist if and only if n ¹ 2p^{a} where p is prime, and n is either 2 or 4 (mod 8).
 WENDY MYRVOLD, University of Victoria, Dept. of Computer Science
Fast Enumeration of All Independent Sets of a Graph
[PDF] 
An independent set of a graph G is a set of vertices of G
which are pairwise nonadjacent. There are many applications for
which the input is a graph G with a large symmetry group and the
goal is to generate either all of the independent sets or all of the
maximum independent sets up to isomorphism. We present a very fast
practical algorithm for this problem. The tactic can also be applied
to many other problems: some examples are generation of all colourings
or matchings of a graph up to isomorphism. Two applications are given
to illustrate the utility of this algorithm: finding all independent
sets of the C60 and C70 fullerenes, and also the problem of finding a
maximum set of vertices at distance four or more in the 120cell.
This is joint work with Patrick Fowler.
 ORTRUD OELLERMANN, University of Winnipeg, 515 Portage Avenue, Winnipeg
The Strong Metric Dimension of a Graph
[PDF] 
Let G be a connected graph. A vertex w is said to strongly
resolve a pair u,v of vertices of G if there exists some shortest
uw path containing v or some shortest vw path containing u.
A set W of vertices is a strong resolving set for G if every pair
of vertices of G is strongly resolved by some vertex of W. A
smallest strong resolving set for G is called a strong basis for G
and its cardinality the strong dimension of G. We begin with a
motivation and an overview of this invariant and a related invariant,
namely the metric dimension of a graph. We then show that the problem
of finding the strong dimension of a connected graph can be
transformed to the problem of finding the vertex covering number of a
graph. Moreover, this invariant is shown to be NPhard.
Joint work with J. PetersFransen.
 DAVID PIKE, Memorial University of Newfoundland, St. John's, Newfoundland
Pancyclic PBD blockintersection graphs
[PDF] 
A pairwise balanced design PBD(v,K,l) consists of
a set V of cardinality v, a set K of positive
integers, and a set B of subsets of V with the properties
that b Î K for each b Î B, and each pair of
elements from V occurs in exactly l of the subsets in
B. The elements of B are known as the blocks of the
design.
Given a combinatorial design D with block set B, its
blockintersection graph G_{D} is the graph having vertex set
B such that two vertices b_{1} and b_{2} are adjacent if and
only if b_{1} and b_{2} have nonempty intersection.
Hare showed in 1995 that if D is a PBD(v,K,1)
with minK \geqslant 3, then G_{D} is
edgepancyclic (i.e., each edge of G_{D} is contained in
a cycle of each length l = 3,4,...,V(G_{D})). In this
presentation we consider blockintersection graphs of pairwise
balanced designs PBD(v,K,l) for which l\geqslant 2.
This is joint work with Graham Case.
 MATEJA SAJNA, University of Ottawa, Department of Mathematics and
Statistics, 585 King Edward Avenue, Ottawa, ON K1N 6N5
Two models for transitive closure of bipolar weighted
digraphs
[PDF] 
A bipolar weighted digraph is a digraph together with a weight
function and a sign function on the arcs such that the weight of each
arc lies in the interval [0,1] and no two parallel arcs have the same
sign. Bipolar weighted digraphs are natural models for socalled
fuzzy cognitive maps, which are used in science, engineering, and the
social sciences to represent a body of knowledge. It has been noted
in the literature that a transitive closure of a bipolar weighted
digraph contains useful new information for the fuzzy cognitive map it
models.
It is natural to define a transitive closure of a bipolar weighted
digraph D=(V,A) as a bipolar weighted digraph D^{*}=(V,A^{*}) such
that an arc (u,v) of sign s is in A^{*} if and only if D has a
directed (u,v)walk of sign s (where the sign of a directed walk
is defined as the product of signs of all its arcs). But what weight
should be assigned to (u,v) in D^{*}? We propose two models: the
bipolar fuzzy digraph model, which has been previously considered in
some form in fuzzy systems research, and the new bipolar random
digraph model. A bipolar fuzzy digraph consists of two fuzzy
relations on the set V (that is, the arc weights are viewed as
degrees of membership), and its transitive closure is a combined
transitive closure of the two fuzzy relations. In a random digraph
model, on the other hand, the arc weights are viewed as probabilities,
and the weight of an arc (u,v) of sign s in the transitive closure
is defined as the probability of having a directed (u,v)walk of
sign s in D. While a version of RoyWarshall's algorithm
efficiently computes the transitive closure of a bipolar fuzzy
digraph, the problem is computationally hard for bipolar random
digraphs. However, we describe several approaches that allow
computation at least for the types and sizes of fuzzy cognitive maps
we have dealt with in practice.
 JOZSEF SOLYMOSI, University of British Columbia, Vancouver
Extremal problems for linear kuniform hypergraphs
[PDF] 
A hypergraph is linear if every two hyperedges share one point at
most. We investigate the following question. What is the maximum
possible number of edges in a linear kuniform hypergraph H with
n vertices which does not contain a given linear kuniform
hypergraph G as a subgraph? We show applications to geometry and
number theory.
 LADISLAV STACHO, Simon Fraser University, Burnaby
Ordered kcolorings dichotomy
[PDF] 
We introduce three variants of proper colorings with imposed partial
ordering on the set of colors and will present dichotomy theorems that
separate these problems into tractable and NPcomplete.
Vertices of all considered graphs are integers from 1 to V(G),
hence they form a linearly ordered set (V (G), £ ). The set of
vertices colored c will be denoted by V_{c}. Given a partially
ordered set (poset) (C,\preceq) of colors, in the first
problem we want to (properly) color vertices of G by colors in
C (color G by poset C) such that for
any two colors c and c¢ if c\preceq c¢ then for any two vertices
u Î V_{c} and v Î V_{c¢}, u £ v. Thus, if \preceq is the
empty relation on C, then the problem is whether G can
be properly colored with C colors, a well known graph
coloring problem.
In the second problem, we want to color G by poset C
such that for any two colors c and c¢ if c\preceq c¢ then for
any two adjacent vertices u Î V_{c} and v Î V_{c¢}, u £ v.
This problem is the wellknown directed graph homomorphism problem
whose dichotomy was extensively studied.
In the last problem, we want to color G by poset C such
that for any two colors c and c¢ if c\preceq c¢ then for any two
vertices u Î V_{c} and v Î V_{c¢} in a component induced by
V_{c}ÈV_{c¢}, u £ v.
This is a joint work with Arvind Gupta, Jan van den Heuvel, Jan
Manuch, and Xiaohong Zhao.
 BAOGANG XU, Nanjing Normal University, Nanjing, 210097, P.R. China
Some results on minimally circularimperfect graphs
[PDF] 
A graph is circularperfect if each of its induced subgraphs has the
same circular chromatic number as its circular clique number. A graph
is called minimally circularimperfect if the graph itself is not
circularperfect but each of its proper induced subgraphs is. One
approach to study the circularperfect graphs is to characterize the
minimally circularimperfect graphs. In this talk, some results on
minimally circularimperfect graphs are presented.

