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### Balanced k-Colorings [chapter]

Therese C. Biedl, Eowyn Cenek, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, Rudolf Fleischer, Ming-Wei Wang
2000 Lecture Notes in Computer Science
While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k ¿ 2, a set of vertices to minimize imbalance among a family of subsets of vertices  ...  For 2-colorings, the bound on the discrepancy is atmost max{2d − 3; 2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. (E.D.  ...  Theorem 1 (Balance Theorem). Let d¿2 and k¿2. Let L be a set of lines of dimension d containing a set of vertices P. Then P has a (4d−3)-balanced k-coloring.  ...

### Balanced k-colorings

Therese C. Biedl, Eowyn Čenek, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, Rudolf Fleischer, Ming-Wei Wang
2002 Discrete Mathematics
While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k ¿ 2, a set of vertices to minimize imbalance among a family of subsets of vertices  ...  For 2-colorings, the bound on the discrepancy is atmost max{2d − 3; 2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. (E.D.  ...  Theorem 1 (Balance Theorem). Let d¿2 and k¿2. Let L be a set of lines of dimension d containing a set of vertices P. Then P has a (4d−3)-balanced k-coloring.  ...

### Colorings of k-balanced matrices and integer decomposition property of related polyhedra

Giacomo Zambelli
2007 Operations Research Letters
This is accomplished by proving certain coloring properties of these matrices.  ...  This concept is related to that of k-balancedness as follows.  ...  The proof of Theorem 1 will follow from certain coloring properties of the class of k-balanced matrices, that we present in the next section.  ...

### Parallel Heuristics for Improved, Balanced Graph Colorings

Robert K. Gjertsen, Jr., Mark T. Jones, Paul E. Plassmann
1996 Journal of Parallel and Distributed Computing
In the second part of the paper we introduce two new parallel color-balancing heuristics, PDR(k) and PLF(k).  ...  We show that these heuristics have the desirable property that they do not increase the number of colors used by an initial coloring during the balancing process.  ...  To achieve the second objective, balanced colorings, we introduced the the PLF(k) and PDR(k) heuristics.  ...

### Performance of the Caltech Submillimeter Observatory Dual-Color 180–720 GHz Balanced SIS Receivers

J. W. Kooi, R. A. Chamberlin, R. Monje, A. Kovacs, F. Rice, H. Yoshida, B. Force, K. Cooper, D. Miller, M. Gould, D. Lis, B. Bumble (+3 others)
2014 IEEE Transactions on Terahertz Science and Technology
We report on balanced SIS receivers covering the astronomical important 180-720 GHz submillimeter atmospheric window.  ...  Instrumental stability is excellent supporting the LO noise cancellation properties of the balanced mixer configuration.  ...  Zmuidzinas of the California Institute of Technology for providing the K a -band synthesizers, the wideband Fast Fourier Transform Spectrometers (FFTS), and for his advise and physics insight over the  ...

### Brightness, Contrast, and Color Balance of Digital versus Film Retinal Images in the Age-Related Eye Disease Study 2

Larry D. Hubbard, Ronald P. Danis, Michael W. Neider, Dennis W. Thayer, Hugh D. Wabers, James K. White, Anthony J. Pugliese, Michael F. Pugliese
2008 Investigative Ophthalmology and Visual Science
Image brightness (B), contrast (C), and color balance (CB) were measured with three-color luminance histograms.  ...  Exemplar images had similar brightness, contrast, and color balance, supporting an image model.  ...  temperature, 6300°K) from the original AREDS.  ...

### Balanced k-decompositions of graphs

Hsiang-Chun Hsu, Gerard Jennhwa Chang
2012 Discrete Applied Mathematics
For a given integer k ≥ 2, a balanced k-coloring of a graph G is a mapping c: In this paper, we determine f k (G) for some graphs of high connectivity, trees and complete multipartite graphs.  ...  Consider a balanced k-coloring c of G that colors tm + 1 vertices of S 1 by i for each color i from 1 to k, and leaves the other vertices uncolored.  ...  In all the above cases, we can choose the other k − 3 color classes A 4 , A 5 , . . . , A k such that all leaves are colored. We claim that this balanced k-coloring is optimal.  ...

### An Optimal Algorithm for Plane Matchings in Multipartite Geometric Graphs [chapter]

Ahmad Biniaz, Anil Maheshwari, Subhas C. Nandy, Michiel Smid
2015 Lecture Notes in Computer Science
Given a color-balanced point set P , a balanced cut is a line which partitions P into two colorbalanced point sets, each of size at most 2n/3+1.  ...  Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most n/2 .  ...  Since P is color-balanced, K n (P 1 , . . . , P k ) has Θ(n 2 ) edges.  ...

### An optimal algorithm for plane matchings in multipartite geometric graphs

Ahmad Biniaz, Anil Maheshwari, Subhas C. Nandy, Michiel Smid
2017 Computational geometry
Given a color-balanced point set P , a balanced cut is a line which partitions P into two colorbalanced point sets, each of size at most 2n/3+1.  ...  Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most n/2 .  ...  Since P is color-balanced, K n (P 1 , . . . , P k ) has Θ(n 2 ) edges.  ...

### Balanced Interval Coloring

Antonios Antoniadis, Falk Hueffner, Pascal Lenzner, Carsten Moldenhauer, Alexander Souza, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science
This problem naturally models a load balancing scenario, where n tasks with given start-and endtimes have to be distributed among k servers. Our results imply that this can be done ideally balanced.  ...  We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal.  ...  any k ∈ ◆, there is a balanced k-coloring.  ...

### Balanced Interval Coloring [article]

Antonios Antoniadis and Falk Hüffner and Pascal Lenzner and Carsten Moldenhauer and Alexander Souza
2010 arXiv   pre-print
This problem naturally models a load balancing scenario, where n tasks with given start- and endtimes have to be distributed among k servers. Our results imply that this can be done ideally balanced.  ...  We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal.  ...  Existence of Balanced k-Colorings We begin by observing the existence of balanced k-colorings.  ...

### Good and nice colorings of balanced hypergraphs

D. de Werra
2006 Discrete Mathematics
Cornuejols, we show that oriented balanced hypergraphs have good k-colorings for any k. A stronger type of coloring is also shown to exist for oriented balanced hypergraphs.  ...  Berge who showed their existence in balanced hypergraphs. There is a simple generalization to oriented balanced hypergraphs: using the bi-coloring characterization of 0, ±1-matrices of M.  ...  . , S k ) be a good k-coloring of an oriented balanced hypergraph H.  ...

### Balanced coloring of bipartite graphs

Uriel Feige, Shimon Kogan
2009 Journal of Graph Theory
A balanced coloring of G is a coloring of the vertices of G such that each color class induces a balanced bipartite independent set in G. If graph G has a balanced coloring we call it colorable.  ...  The coloring number χ B (G) is the minimum number of colors in a balanced coloring of a colorable graph G.  ...  Given a balanced bipartite independent set I of size k in G r one can extend the r-partial coloring into a (r − k)-partial coloring in the following manner.  ...

### Colored unavoidable patterns and balanceable graphs [article]

Matt Bowen, Adriana Hansberg, Amanda Montejano, Alp Müyesser
2021 arXiv   pre-print
We call an edge-coloring of a path P_rk balanced if each color appears k times in the coloring.  ...  We show that any 3-edge-coloring of a large complete graph with kn+o(n) edges in each color contains a balanced P_3k. This is tight up to a constant factor of 2.  ...  Further, we say that an r-coloring of E(K n ) contains a balanced copy of G if it admits a balanced embedding of G.  ...

### Balance in Random Trees

Azer Akhmedov, Warren Shreve
2014 Open Journal of Discrete Mathematics
We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly k -balanced for any 3 k ≥ .  ...  Edges with different colored endpoints are left uncolored. G is said to be balanced if neither the number of vertices nor and the number of edges of the two different colors differs by more than one.  ...  The map c will be called a k -balanced coloring. Definition 1.5 Let 2 k ≥ .  ...
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