scholarly journals DICHROMATIC NUMBER AND FRACTIONAL CHROMATIC NUMBER

2016 ◽  
Vol 4 ◽  
Author(s):  
BOJAN MOHAR ◽  
HEHUI WU
Keyword(s):  
Chromatic Number ◽  
Constant Factor ◽  
Independent Interest ◽  
Directed Cycle ◽  
Significant Progress ◽  
Fractional Chromatic Number ◽  
Small Constant ◽  
Dichromatic Number

The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and Neumann-Lara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We make the first significant progress on this conjecture by proving a fractional version of the conjecture. While our result uses a stronger assumption about the fractional chromatic number, it also gives a much stronger conclusion: if the fractional chromatic number of a graph is at least $t$, then the fractional version of the dichromatic number of the graph is at least ${\textstyle \frac{1}{4}}t/\log _{2}(2et^{2})$. This bound is best possible up to a small constant factor. Several related results of independent interest are given.

scholarly journals Complete Acyclic Colorings

10.37236/8752 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Stefan Felsner ◽  
Winfried Hochstättler ◽  
Kolja Knauer ◽  
Raphael Steiner
Keyword(s):  
Chromatic Number ◽  
Undirected Graph ◽  
Directed Cycle ◽  
Vertex Arboricity ◽  
Classical Parameters ◽  
Two Parameters ◽  
Vertex Sets ◽  
Dichromatic Number

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.

scholarly journals Cubical coloring — fractional covering by cuts and semidefinite programming

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Robert Šámal
Keyword(s):  
Semidefinite Programming ◽  
Chromatic Number ◽  
Fractional Chromatic Number ◽  
Maximum Cut ◽  
Continuous Mappings ◽  
Polynomial Time Approximation Algorithm ◽  
Eigenvalue Bound ◽  
International Audience ◽  
Graph Parameters ◽  
Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

SPARSE PARTITION REGULARITY

2006 ◽  
Vol 93 (3) ◽  
pp. 545-569 ◽  
Author(s):  
IMRE LEADER ◽  
PAUL A. RUSSELL
Keyword(s):  
Chromatic Number ◽  
Independent Interest ◽  
Natural Numbers ◽  
Ramsey Theorem

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

THE MATCHED FIELD PROCESSING OF BINAURAL DOLPHIN-LIKE SIGNALS FOR THE DETECTION AND IDENTIFICATION OF BURIED TARGETS

2003 ◽  
Vol 11 (04) ◽  
pp. 521-534 ◽  
Author(s):  
A. TOLSTOY ◽  
W. AU
Keyword(s):  
Real Data ◽  
Constant Factor ◽  
Data Display ◽  
Signal To Noise ◽  
Matched Field Processing ◽  
Detection And Identification ◽  
Small Constant ◽  
Target Template ◽  
New Type ◽  
Target Data

The Matched Field Processing (MFP) approach to be discussed here is intended to extract subtle differences between apparently similar signals. The technique is applied coherently to an array of data, i.e. to two receivers. One of the main advantages to this work is that even though we use MFP, there is no modeling involved. Since the available binaural data are quite limited and show very strong, obviously different returns from all the targets (not the subtle differences realistically expected), we found it necessary to manipulate the data to bring them more into line with expectations. In particular, scattered returns from a drum were reduced, i.e. multiplied by a small constant factor, then added to the scattered returns from bottom-only data using various time shifts. The shifts simulated a family of returns from a low signal-to-noise (S/N) 55 gallon drum target. This family with shifted bottom scattering mimics returns from multiple placements of the targets on the bottom. These new target "data" (comprised of manipulated real data) seem at first glance to be nearly identical to the original bottom-only returns. Thus, the new target data display subtle differences from the bottom-only data. The MFP approach (based on the linear, a.k.a., Bartlett, processor) was then applied to these new "data". They were processed and yielded a target "template" of scattered returns varying as a function of time and frequency characterizing the returns scattered from the drum. Additionally, a similar template was computed for the buried manta-like target data and is seen to be quite different from the drum template. This new type of template can easily be used to detect scattering from particular target types in low S/N situations. It is not proposed that dolphins are using these templates, but, rather, that the templates display scattering characteristics which the dolphins may be using. More data would be extremely useful in determining the templates under a variety of conditions, e.g. for lower S/N levels, different bottom types, targets types, source ranges, depths, and scattering angles, etc.

scholarly journals Precision-Recall versus Accuracy and the Role of Large Data Sets

2019 ◽  
Vol 33 ◽  
pp. 4039-4048 ◽  
Author(s):  
Brendan Juba ◽  
Hai S. Le
Keyword(s):  
Machine Learning ◽  
Class Imbalance ◽  
Imbalanced Data ◽  
Large Data ◽  
Constant Factor ◽  
Data Sets ◽  
Data Set ◽  
Small Constant ◽  
Classifier Performance ◽  
Necessary And Sufficient

Practitioners of data mining and machine learning have long observed that the imbalance of classes in a data set negatively impacts the quality of classifiers trained on that data. Numerous techniques for coping with such imbalances have been proposed, but nearly all lack any theoretical grounding. By contrast, the standard theoretical analysis of machine learning admits no dependence on the imbalance of classes at all. The basic theorems of statistical learning establish the number of examples needed to estimate the accuracy of a classifier as a function of its complexity (VC-dimension) and the confidence desired; the class imbalance does not enter these formulas anywhere. In this work, we consider the measures of classifier performance in terms of precision and recall, a measure that is widely suggested as more appropriate to the classification of imbalanced data. We observe that whenever the precision is moderately large, the worse of the precision and recall is within a small constant factor of the accuracy weighted by the class imbalance. A corollary of this observation is that a larger number of examples is necessary and sufficient to address class imbalance, a finding we also illustrate empirically.

A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
Author(s):  
VAN H. VU
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Present Author ◽  
Upper Bounds ◽  
Constant Factor ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Critical Graphs ◽  
Multiplicative Constant ◽  
List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

A NEAR-OPTIMAL BROADCASTING PROTOCOL FOR MOBILE VIDEO-ON-DEMAND

2009 ◽  
Vol 20 (01) ◽  
pp. 45-55
Author(s):  
REGANT Y. S. HUNG ◽  
H. F. TING
Keyword(s):  
Video On Demand ◽  
Constant Factor ◽  
Mobile Video ◽  
Bandwidth Requirement ◽  
On Demand ◽  
Small Constant ◽  
Wide Range ◽  
New Type ◽  
Broadcasting Protocol ◽  
Mobile Clients

The advance of wireless and mobile technology introduces a new type of Video-on-Demand (VOD) systems, namely the mobile VOD systems, that provide VOD services to mobile clients. It is a challenge to design broadcasting protocols for such systems because of the following special requirements: (1) fixed maximum bandwidth requirement: the maximum bandwidth required for broadcasting a movie should be fixed and independent of the number of requests, (2) load adaptivity: the total bandwidth usage should be dependent on the number of requests; the fewer the requests the smaller the total bandwidth usage, and (3) clients sensitivity: the system should be able to support clients with a wide range of heterogeneous capabilities. In the literature, there are some partial solutions that give protocols meeting one or two of the above requirements. In this paper, we give the first protocol that meets all of the three requirements. The performance of our protocol is optimal up to a small constant factor.

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