3,402 research outputs found
Tiling, spectrality and aperiodicity of connected sets
Let be a set of finite measure. The periodic
tiling conjecture suggests that if tiles by
translations then it admits at least one periodic tiling. Fuglede's conjecture
suggests that admits an orthogonal basis of exponential functions if
and only if it tiles by translations. Both conjectures are known
to be false in sufficiently high dimensions, with all the so-far-known
counterexamples being highly disconnected. On the other hand, both conjectures
are known to be true for convex sets. In this work we study these conjectures
for connected sets. We show that the periodic tiling conjecture, as well as
both directions of Fuglede's conjecture are false for connected sets in
sufficiently high dimensions.Comment: 20 pages, 8 figure
Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs
In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero
5-flow. Let be the minimum number of odd cycles in a 2-factor of a
bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to
cubic graphs with . We show that if a cubic graph has no
edge cut with fewer than edges that separates two odd
cycles of a minimum 2-factor of , then has a nowhere-zero 5-flow. This
implies that if a cubic graph is cyclically -edge connected and , then has a nowhere-zero 5-flow
On a conjecture of Brouwer involving the connectivity of strongly regular graphs
In this paper, we study a conjecture of Andries E. Brouwer from 1996
regarding the minimum number of vertices of a strongly regular graph whose
removal disconnects the graph into non-singleton components.
We show that strongly regular graphs constructed from copolar spaces and from
the more general spaces called -spaces are counterexamples to Brouwer's
Conjecture. Using J.I. Hall's characterization of finite reduced copolar
spaces, we find that the triangular graphs , the symplectic graphs
over the field (for any prime power), and the
strongly regular graphs constructed from the hyperbolic quadrics
and from the elliptic quadrics over the field ,
respectively, are counterexamples to Brouwer's Conjecture. For each of these
graphs, we determine precisely the minimum number of vertices whose removal
disconnects the graph into non-singleton components. While we are not aware of
an analogue of Hall's characterization theorem for -spaces, we show
that complements of the point graphs of certain finite generalized quadrangles
are point graphs of -spaces and thus, yield other counterexamples to
Brouwer's Conjecture.
We prove that Brouwer's Conjecture is true for many families of strongly
regular graphs including the conference graphs, the generalized quadrangles
graphs, the lattice graphs, the Latin square graphs, the strongly
regular graphs with smallest eigenvalue -2 (except the triangular graphs) and
the primitive strongly regular graphs with at most 30 vertices except for few
cases.
We leave as an open problem determining the best general lower bound for the
minimum size of a disconnecting set of vertices of a strongly regular graph,
whose removal disconnects the graph into non-singleton components.Comment: 25 pages, 1 table; accepted to JCTA; revised version contains a new
section on copolar and Delta space
Constructing highly arc transitive digraphs using a layerwise direct product
We introduce a construction of highly arc transitive digraphs using a
layerwise direct product. This product generalizes some known classes of highly
arc transitive digraphs but also allows to construct new such. We use the
product to obtain counterexamples to a conjecture by Cameron, Praeger and
Wormald on the structure of certain highly arc transitive digraphs.Comment: 16 page
Concordance of actions on
We consider locally linear Z_p x Z_p actions on the four-sphere. We present
simple constructions of interesting examples, and then prove that a given
action is concordant to its linear model if and only if a single surgery
obstruction taking to form of an Arf invariant vanishes. We discuss the
behavior of this invariant under various connected-sum operations, and conclude
with a brief discussion of the existence of actions which are not concordant to
their linear models
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