660 research outputs found
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
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
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
THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References
and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a
Revisiting path-type covering and partitioning problems
This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts
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