925 research outputs found
Perfect countably infinite Steiner triple systems
We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2ℵ0 non-isomorphic perfect systems
Inferring an Indeterminate String from a Prefix Graph
An \itbf{indeterminate string} (or, more simply, just a \itbf{string}) \s{x}
= \s{x}[1..n] on an alphabet is a sequence of nonempty subsets of
. We say that \s{x}[i_1] and \s{x}[i_2] \itbf{match} (written
\s{x}[i_1] \match \s{x}[i_2]) if and only if \s{x}[i_1] \cap \s{x}[i_2] \ne
\emptyset. A \itbf{feasible array} is an array \s{y} = \s{y}[1..n] of
integers such that \s{y}[1] = n and for every , \s{y}[i] \in
0..n\- i\+ 1. A \itbf{prefix table} of a string \s{x} is an array \s{\pi} =
\s{\pi}[1..n] of integers such that, for every , \s{\pi}[i] = j
if and only if \s{x}[i..i\+ j\- 1] is the longest substring at position
of \s{x} that matches a prefix of \s{x}. It is known from \cite{CRSW13} that
every feasible array is a prefix table of some indetermintate string. A
\itbf{prefix graph} \mathcal{P} = \mathcal{P}_{\s{y}} is a labelled simple
graph whose structure is determined by a feasible array \s{y}. In this paper we
show, given a feasible array \s{y}, how to use \mathcal{P}_{\s{y}} to
construct a lexicographically least indeterminate string on a minimum alphabet
whose prefix table \s{\pi} = \s{y}.Comment: 13 pages, 1 figur
Cyclic Critical Groups of Graphs
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly
Vertex arboricity of triangle-free graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
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