438 research outputs found
On equitably 2-colourable odd cycle decompositions
An -cycle decomposition of is said to be \emph{equitably
-colourable} if there is a -vertex-colouring of such that each
colour is represented (approximately) an equal number of times on each cycle:
more precisely, we ask that in each cycle of the decomposition, each colour
appears on or of the vertices
of . In this paper we study the existence of equitably 2-colourable
-cycle decompositions of , where is odd, and prove the
existence of such a decomposition for (mod ).Comment: 24p
Social Media Days at UMass Boston
Hosted by Professor Werner Kunz, Social Media Days is envisioned to be a meeting place and networking hub for Boston businesses and organizations interested in Social Media. This daylong event combines presentations from high profile speakers with breakout discussions/small group workshops. Attendees can expect high quality and knowledgeable speakers and an increased amount of face to face interaction. Social Media Days strengthens the connection between UMass Boston and the local business community through an engaging day long event
Generalized packing designs
Generalized -designs, which form a common generalization of objects such
as -designs, resolvable designs and orthogonal arrays, were defined by
Cameron [P.J. Cameron, A generalisation of -designs, \emph{Discrete Math.}\
{\bf 309} (2009), 4835--4842]. In this paper, we define a related class of
combinatorial designs which simultaneously generalize packing designs and
packing arrays. We describe the sometimes surprising connections which these
generalized designs have with various known classes of combinatorial designs,
including Howell designs, partial Latin squares and several classes of triple
systems, and also concepts such as resolvability and block colouring of
ordinary designs and packings, and orthogonal resolutions and colourings.
Moreover, we derive bounds on the size of a generalized packing design and
construct optimal generalized packings in certain cases. In particular, we
provide methods for constructing maximum generalized packings with and
block size or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd
British Combinatorial Conference, July 201
A survey on constructive methods for the Oberwolfach problem and its variants
The generalized Oberwolfach problem asks for a decomposition of a graph
into specified 2-regular spanning subgraphs , called factors.
The classic Oberwolfach problem corresponds to the case when all of the factors
are pairwise isomorphic, and is the complete graph of odd order or the
complete graph of even order with the edges of a -factor removed. When there
are two possible factor types, it is called the Hamilton-Waterloo problem.
In this paper we present a survey of constructive methods which have allowed
recent progress in this area. Specifically, we consider blow-up type
constructions, particularly as applied to the case when each factor consists of
cycles of the same length. We consider the case when the factors are all
bipartite (and hence consist of even cycles) and a method for using circulant
graphs to find solutions. We also consider constructions which yield solutions
with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series.
23 pages, 2 figure
Existential Closure in Line Graphs
A graph is -existentially closed if, for all disjoint sets of vertices
and with , there is a vertex not in adjacent
to each vertex of and to no vertex of .
In this paper, we investigate -existentially closed line graphs. In
particular, we present necessary conditions for the existence of such graphs as
well as constructions for finding infinite families of such graphs. We also
prove that there are exactly two -existentially closed planar line graphs.
We then consider the existential closure of the line graphs of hypergraphs and
present constructions for -existentially closed line graphs of hypergraphs.Comment: 13 pages, 2 figure
The Edge-Connectivity of Vertex-Transitive Hypergraphs
A graph or hypergraph is said to be vertex-transitive if its automorphism
group acts transitively upon its vertices. A classic theorem of Mader asserts
that every connected vertex-transitive graph is maximally edge-connected. We
generalise this result to hypergraphs and show that every connected linear
uniform vertex-transitive hypergraph is maximally edge-connected. We also show
that if we relax either the linear or uniform conditions in this
generalisation, then we can construct examples of vertex-transitive hypergraphs
which are not maximally edge-connected.Comment: 8 page
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