6,985 research outputs found
Restricted Size Ramsey Number for Path of Order Three Versus Graph of Order Five
Let and be simple graphs. The Ramsey number for a pair of graph and is the smallest number such that any red-blue coloring of edges of contains a red subgraph or a blue subgraph . The size Ramsey number for a pair of graph and is the smallest number such that there exists a graph with size satisfying the property that any red-blue coloring of edges of contains a red subgraph or a blue subgraph . Additionally, if the order of in the size Ramsey number is , then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller\u27s works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five
Restricted Size Ramsey Number for Matching versus Tree and Triangle Unicyclic Graphs of Order Six
Let F, G, and H be simple graphs. The graph F arrows (G,H) if for any red-blue coloring on the edge of F, we find either a red-colored graph G or a blue-colored graph H in F. The Ramsey number r(G,H) is the smallest positive integer r such that a complete graph Kr arrows (G,H). The restricted size Ramsey number r∗(G,H) is the smallest positive integer r∗ such that there is a graph F, of order r(G,H) and with the size r∗, satisfying F arrows (G,H). In this paper we give the restricted size Ramsey number for a matching of two edges versus tree and triangle unicyclic graphs of order six
Online Ramsey numbers: Long versus short cycles
Online Ramsey game is played between Builder and Painter on an infinite board
. In every round Builder selects an edge, then Painter colors it
red or blue. Both know target graphs and . Builder aims to create
either a red copy of or a blue copy of in as soon
as possible, and Painter tries to prevent it. The online Ramsey number
is the minimum number of rounds such that the Builder
wins. We study where is fixed and is large. We
show that for an absolute constant if
is even, while if is odd
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
The DLV System for Knowledge Representation and Reasoning
This paper presents the DLV system, which is widely considered the
state-of-the-art implementation of disjunctive logic programming, and addresses
several aspects. As for problem solving, we provide a formal definition of its
kernel language, function-free disjunctive logic programs (also known as
disjunctive datalog), extended by weak constraints, which are a powerful tool
to express optimization problems. We then illustrate the usage of DLV as a tool
for knowledge representation and reasoning, describing a new declarative
programming methodology which allows one to encode complex problems (up to
-complete problems) in a declarative fashion. On the foundational
side, we provide a detailed analysis of the computational complexity of the
language of DLV, and by deriving new complexity results we chart a complete
picture of the complexity of this language and important fragments thereof.
Furthermore, we illustrate the general architecture of the DLV system which
has been influenced by these results. As for applications, we overview
application front-ends which have been developed on top of DLV to solve
specific knowledge representation tasks, and we briefly describe the main
international projects investigating the potential of the system for industrial
exploitation. Finally, we report about thorough experimentation and
benchmarking, which has been carried out to assess the efficiency of the
system. The experimental results confirm the solidity of DLV and highlight its
potential for emerging application areas like knowledge management and
information integration.Comment: 56 pages, 9 figures, 6 table
Causal Discovery with Continuous Additive Noise Models
We consider the problem of learning causal directed acyclic graphs from an
observational joint distribution. One can use these graphs to predict the
outcome of interventional experiments, from which data are often not available.
We show that if the observational distribution follows a structural equation
model with an additive noise structure, the directed acyclic graph becomes
identifiable from the distribution under mild conditions. This constitutes an
interesting alternative to traditional methods that assume faithfulness and
identify only the Markov equivalence class of the graph, thus leaving some
edges undirected. We provide practical algorithms for finitely many samples,
RESIT (Regression with Subsequent Independence Test) and two methods based on
an independence score. We prove that RESIT is correct in the population setting
and provide an empirical evaluation
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