Relevant bibliographies by topics / The two color theorem

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'The two color theorem'

Author: Grafiati
Published: 2 June 2025
Last updated: 31 July 2025

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Journal articles on the topic "The two color theorem"

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Doi, H. "The "four-color issue" in ecology for considering ecosystem boundaries." Web Ecology 13, no. 1 (2013): 91–93. http://dx.doi.org/10.5194/we-13-91-2013.

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Abstract. Ecosystem boundaries are important structures in defining ecosystems. To date, ecologists have not extensively considered which boundaries are important in explaining ecological phenomena in order to simplify ecological theories. The four-color theorem in mathematics maintains that only four colors are required to color a set of regions so that no two adjacent regions have the same color. Before being proven in 1976, the theorem was considered the "four-color issue", which proposed that a small number of colors were required to separate regional boundaries. Applying the principle of
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White, Jacob Anthony. "Brooks' theorem for 2-fold coloring." Contributions to Discrete Mathematics 17, no. 2 (2022): 67–76. http://dx.doi.org/10.55016/ojs/cdm.v17i2.62707.

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The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph. We prove that, for 2-connected graphs, the two-fold chromatic number is at most the Ore degree, unless G is a complete graph or an odd cycle.
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Goldstein, Larry, and Yosef Rinott. "Multivariate normal approximations by Stein's method and size bias couplings." Journal of Applied Probability 33, no. 1 (1996): 1–17. http://dx.doi.org/10.2307/3215259.

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Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given de
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Goldstein, Larry, and Yosef Rinott. "Multivariate normal approximations by Stein's method and size bias couplings." Journal of Applied Probability 33, no. 01 (1996): 1–17. http://dx.doi.org/10.1017/s0021900200103675.

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Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given de
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Aletti, Giacomo, Caterina May, and Piercesare Secchi. "A central limit theorem, and related results, for a two-color randomly reinforced urn." Advances in Applied Probability 41, no. 3 (2009): 829–44. http://dx.doi.org/10.1239/aap/1253281065.

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We prove a central limit theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.
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Aletti, Giacomo, Caterina May, and Piercesare Secchi. "A central limit theorem, and related results, for a two-color randomly reinforced urn." Advances in Applied Probability 41, no. 03 (2009): 829–44. http://dx.doi.org/10.1017/s0001867800003578.

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We prove a central limit theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.
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Shen, Zhe, Yudong Ma, and Takeshi Tsuchiya. "Four-Dimensional Gait Surfaces for a Tilt-Rotor—Two Color Map Theorem." Drones 6, no. 5 (2022): 103. http://dx.doi.org/10.3390/drones6050103.

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This article presents the four-dimensional surfaces that guide the gait plan for a tilt-rotor. The previous gaits analyzed in the tilt-rotor research are inspired by animals; no theoretical base backs the robustness of these gaits. This research deduces the gaits by diminishing the adverse effect of the attitude of the tilt-rotor for the first time. Four-dimensional gait surfaces are subsequently found on which the gaits are expected to be robust to the attitude. These surfaces provide the region where the gait is suggested to be planned. However, a discontinuous region may hinder the gait pla
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Liang, Zuosong, and Huandi Wei. "A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs." Computational Intelligence and Neuroscience 2021 (October 5, 2021): 1–5. http://dx.doi.org/10.1155/2021/7667656.

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Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar
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Narendra, Risang. "Polinomial Kromatik Graf Bunga." Unisda Journal of Mathematics and Computer Science (UJMC) 8, no. 2 (2022): 55–58. http://dx.doi.org/10.52166/ujmc.v8i2.3713.

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In this study, the class of a graph called Flower Graph was discussed. Flower Graph is a special way in a graphic. A graph G is called a flower chart-. This graph will be symbolized by . Then, it is defined by a flower chart-, with the petal to which in a flower graph-, by -petal removed, for . Flower graphics-: with petal as much as is denoted by . Chromatic polynomial is the amount of forms to color the point in graph G with color, where there are no two points that fit to obtain the same color. In the end, using the reduction theorem, the chromatic polynomial theorem of the Sikel chart and
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Bhamare, Dr Megha Abhiman. "CHROMATIC NUMBER OPTIMIZATION FOR EFFICIENT MAP COLORING OF MAHARASHTRA'S DISTRICTS." International Journal of Engineering Applied Sciences and Technology 09, no. 06 (2024): 153–57. https://doi.org/10.33564/ijeast.2024.v09i06.020.

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This research explores the application of graph coloring algorithms to minimize the number of colors required for coloring the districts of Maharashtra, ensuring that no two adjacent districts share the same color. Using an adjacency matrix representation, where each district corresponds to a vertex and edges represent shared borders, the problem is framed as a graph coloring challenge. The primary objective is to demonstrate that a minimum of four colors suffice to color the entire map, in line with the Four Color Theorem, which asserts that any planar map can be colored with at most four col
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Dissertations / Theses on the topic "The two color theorem"

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Mikhail, Sarah Shawky. "Optical Properties of Two Brands of Composite Restorative Materials and Confirmation of Theoretical Predictions for Layering." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1316436113.

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Leward, Oscar. "Graph TheoryThe Four Color Theorem." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-232809.

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Miller, Shannon J. "Two Approaches to Clifford's Theorem." Youngstown State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1619792031825711.

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Man, Shing Hing. "Two extensions of Milnor's patching theorem." Thesis, Imperial College London, 1990. http://hdl.handle.net/10044/1/46429.

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Nieh, Ari. "Fractional Analogues in Graph Theory." Scholarship @ Claremont, 2001. https://scholarship.claremont.edu/hmc_theses/131.

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Tait showed in 1878 that the Four Color Theorem is equivalent to being able to three-color the edges of any planar, three-regular, two-edge connected graph. Not surprisingly, this equivalent problem proved to be equally difficult. We consider the problem of fractional colorings, which resemble ordinary colorings but allow for some degree of cheating. Happily, it is known that every planar three-regular, two-edge connected graph is fractionally three-edge colorable. Is there an analogue to Tait’s Theorem which would allow us to derive the Fractional Four Color Theorem from this edge-coloring re
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Kundu, Joydip 1977. "Two key questions about color superconductivity." Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/29452.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2004.<br>Includes bibliographical references (p. 109-116).<br>We pose two key questions about color superconductivity: What are the effects of the large strange quark mass, and what are the observable consequences of color superconductivity? Motivated by the first question, we study crystalline color superconductivity. We adapt the Nambu-Gor'kov formalism to study this phase, and go on to examine distinctions between crystalline color superconductivity induced by quark mass differences and by quark chemical potential dif
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MAMELI, VALENTINA. "Two generalizations of the skew-normal distribution and two variants of McCarthy's theorem." Doctoral thesis, Università degli Studi di Cagliari, 2012. http://hdl.handle.net/11584/266181.

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The thesis is structured into two main parts. The first and major part is concerned with the skew-normal distribution, introduced by Azzalini (1985) [6], while the second one is connected with the scoring rules. In part one the problem of finding confidence intervals for the skewness parameter of the skew-normal distribution is addressed. Two new five-parameter continuous distributions which generalize the skew-normal distribution as well as some other well-known distributions are proposed and studied. Some mathematical properties of both distributions are derived. Part two is focused on the e
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Oldfield, J. P. "Two-term Szegő theorem for generalised anti-Wick operators." Thesis, University College London (University of London), 2015. http://discovery.ucl.ac.uk/1463831/.

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This thesis concerns operators whose Weyl pseudodifferential operator symbol is the convolution of a function that is smooth and of fixed scale with a function that is discontinuous and dilated by a large asymptotic parameter. A special case of these operators of particular interest is the class of generalised anti-Wick operators, and then the fixed-scale part corresponds to the window functions while the dilated part is the generalised anti-Wick symbol. The main result is a Szegő theorem that gives two terms in the asymptotic expansion of the trace of a function of the operator. Two variants
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Hoehner, Steven D. "The Hasse-Minkowski Theorem in Two and Three Variables." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338317481.

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Felisetti, Camilla <1990&gt. "Two applications of the decomposition theorem to moduli spaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amsdottorato.unibo.it/8681/1/felisetti_camilla_tesi.pdf.

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The decomposition theorem is a statement about the (derived) direct image of the intersection cohomology by an algebraic projective map. The decomposition theorem and more generally the theory of perverse sheaves have found many interesting applications, especially in representation theory. Usually a lot of work is needed to apply it in concrete situations, to identify the various summands. This thesis proposes two applications of the decomposition theorem. In the first we consider the moduli space of Higgs bundles of rank 2 and degree 0 over a curve of genus 2. Applying the decomposition the
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Books on the topic "The two color theorem"

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Fritsch, Rudolf, and Gerda Fritsch. The Four-Color Theorem. Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1720-6.

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Pinsky, Marina. Marina Pinsky: Four color theorem. Triangle Books, 2021.

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Field, Dahlstrom Carol, ed. Two-color quilts. Meredith Books, 2004.

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Inc, Pantone, ed. Pantone two color selector. Pantone, Inc., 1985.

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Binns, Betty. Designing with two colors. Watson-Guptill Publications, 1991.

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Rudolf, Fritsch. The four color theorem: History, topological foundations, and idea of proof. Springer, 1998.

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McGuire, Hugh W. Two methods for checking formulas of temporal logic. Dept. of Computer Science, Stanford University, 1995.

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S, Weigend Andreas, and Ames Research Center, eds. Two papers on feed-forward networks. NASA Ames Research Center, 1991.

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S, Weigend Andreas, and Ames Research Center, eds. Two papers on feed-forward networks. NASA Ames Research Center, 1991.

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L, Tuma Margaret, and United States. National Aeronautics and Space Administration., eds. Two-color laser speckle shift strain measurement system. National Aeronautics and Space Administration, 1996.

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Book chapters on the topic "The two color theorem"

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Secco, Gisele Dalva, and Luiz Carlos Pereira. "Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem." In How Colours Matter to Philosophy. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-67398-1_17.

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Jia, Zhi-Gang, Si-Tao Ling, and Mei-Xiang Zhao. "Color Two-Dimensional Principal Component Analysis for Face Recognition Based on Quaternion Model." In Intelligent Computing Theories and Application. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-63309-1_17.

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Xu, Jin. "Construction of Maximal Planar Graphs with the Same Order." In Maximal Planar Graph Theory and the Four-Color Conjecture. Springer Nature Singapore, 2025. https://doi.org/10.1007/978-981-96-4745-3_4.

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Abstract From this chapter, we will present a series of methods to construct maximal planar graphs. In this chapter, we first introduce a simple but useful operation, called diagonal flip (or edge-flipping), by a finite sequence of which any two maximal planar graphs with the same order can be transformed into each other.
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Weik, Martin H. "four-color theorem." In Computer Science and Communications Dictionary. Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_7504.

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Soifer, Alexander. "The Four-Color Theorem." In The New Mathematical Coloring Book. Springer US, 2024. http://dx.doi.org/10.1007/978-1-0716-3597-1_22.

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Soifer, Alexander. "The Four-Color Theorem." In The Mathematical Coloring Book. Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-74642-5_21.

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Ben-Ari, Mordechai. "The Five-Color Theorem." In Mathematical Surprises. Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-13566-8_4.

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AbstractMaps use colors to distinguish one region from another by ensuring that adjacent regions are colored with different colors. In 1852 Francis Guthrie noticed that a map of the counties of England could be colored using only four countries. The claim that four countries suffice to color any planar map is called the four-color theorem and was only proved in 1976 by Kenneth Appel andWolfgang Haken. They used sophisticated mathematical arguments to show that if there is a counterexample (a map needing more than four colors), it had to be associated with one of 1834 configurations. They then
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Dineen, Seán. "Green’s Theorem." In Functions of Two Variables. Springer US, 1995. http://dx.doi.org/10.1007/978-1-4899-3250-1_22.

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Fritsch, Rudolf, and Gerda Fritsch. "History." In The Four-Color Theorem. Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1720-6_1.

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Fritsch, Rudolf, and Gerda Fritsch. "(Topological) Maps." In The Four-Color Theorem. Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1720-6_2.

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Conference papers on the topic "The two color theorem"

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Kwok, Ho Pai, Shinichiro Akiyama, and Synge Todo. "Two-color lattice QCD in (1 + 1) dimensions with Grassmann tensor renormalization group." In The 41st International Symposium on Lattice Field Theory. Sissa Medialab, 2025. https://doi.org/10.22323/1.466.0364.

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Chiba, Masato, and Masashi Yamada. "A Perceptual Study on the Ratios of Areas of Two Adjacent Colors for the Optimal Congruency." In 14th International Conference on Applied Human Factors and Ergonomics (AHFE 2023). AHFE International, 2023. http://dx.doi.org/10.54941/ahfe1002995.

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Cute female characters frequently appear in Cool Japan content, such as anime, video games and manga. When designing a character, color is one of the most important factors. Previous studies showed that yellow characters tended to have a cute connotation like Pokémon. A character with yellow hair also had a cute impression. Particularly, a character with yellow hair and blue irises was perceived as one of the cutest types of character. A pair of yellow and blue consists of complementary colors and generally it is said that a pair of complementary colors gives an uncomfortable impression. It co
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Hands, Simon, Seyong Kim, and Jon-Ivar Skullerud. "Two color QCD beyond the BEC regime." In XXIIIrd International Symposium on Lattice Field Theory. Sissa Medialab, 2005. http://dx.doi.org/10.22323/1.020.0149.

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Mulligan, Jeffrey B. "Linear model of color transparency." In OSA Annual Meeting. Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.fdd5.

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When two lights are combined additively, the color that results can be determined from the colors of the two component lights without knowledge of the nature of the actual spectra. This is not the case for a subtractive mixture, which occurs with color transparencies and colored shadows, because the spectrum of the resultant is the product of the two component spectra. Because the actual spectra cannot be known solely from colorimetric data, prediction of the subtractive mixture is an ill-posed problem. Concrete predictions can be made by adopting one of the assumptions of Maloney and Wandell'
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Giudice, Pietro, and Simon Hands. "Two color matter in the quenched approximation." In The XXV International Symposium on Lattice Field Theory. Sissa Medialab, 2008. http://dx.doi.org/10.22323/1.042.0186.

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Kotov, Andrey Yuryevich, Ernst-Michael Ilgenfritz, Victor Braguta, Michael Muller-Preussker, Bengt Petersson, and Alexander Schreiber. "Two-color QCD with chiral chemical potential." In The 32nd International Symposium on Lattice Field Theory. Sissa Medialab, 2015. http://dx.doi.org/10.22323/1.214.0235.

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Krantz, David H. "Nonsingularity of asymmetric color matches: implications for color theory." In OSA Annual Meeting. Optica Publishing Group, 1985. http://dx.doi.org/10.1364/oam.1985.wm1.

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Central mechanisms of color are defined to be the central neural processes that are equated in asymmetric color matching. For normal trichromats, asymmetric color matches are nonsingular: given any match, all sufficiently small perturbations of one of the matched colors can be matched by perturbations of the other. This fact, together with the other laws of color matching, yields a rigorous representation of normal color percepts as a 3-D manifold, and also allows the deduction that all effects of adaptation and contrast can be understood as effects on exactly three independent central mechani
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Zaidi, Qasim, and Arthur G. Shapiro. "Combination of Signals from Opponent Color Mechanisms." In Advances in Color Vision. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/acv.1992.sac4.

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Despite the long standing interest in stage-theories of human color vision, little thought had been given to the processing of the output of traditional opponent mechanisms until Krauskopf et al (1986a,b) presented the first experiments to address this issue. The original evidence for post-opponent mechanisms came from the habituating effect of prolonged simultaneous stimulation of the two chromatic cardinal mechanisms of Krauskopf et al (1982). Under the hypothesis that prolonged stimulation leads to independent "fatigue" of the two mechanisms, the effect of habituation should be greatest alo
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Martinez-Uriegas, Eugenio. "Measurements of afterimages and minimum grating." In Advances in Color Vision. Optica Publishing Group, 1992. http://dx.doi.org/10.1364/acv.1992.sad3.

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Zone theories of the Muller type are major frameworks for psychophysical data on color vision. According to quantitative opponent-color models derived from zone theories, three degrees of freedom given by trichromacy of human photopigments translate into three visual channels. One achromatic or broad-band channel and two color-opponent or chromatic channels. One of the limitations of these models is the absence of explicit spatiotemporal variables in their formulation, in consequence they apply only to data obtained with steady-state aperture colors. Fortunately, a very active area of research
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Shen, Zhe, and Takeshi Tsuchiya. "The Robust Gait of a Tilt-rotor and Its Application to Tracking Control – Application of Two Color Map Theorem." In 2022 22nd International Conference on Control, Automation and Systems (ICCAS). IEEE, 2022. http://dx.doi.org/10.23919/iccas55662.2022.10003826.

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Reports on the topic "The two color theorem"

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Tran, Tan Phuc. Preservation/Restoration 101 - Some Theoretical Perspectives. Mytho International and Philosophical Review, 2023. http://dx.doi.org/10.56624/mtu0001.

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On April 12, 2023, photos of a newly restored French villa in Hanoi with a budget of 14 billion VND were published. Discussions were rigorous with the participation of the public and architects about the seemingly new and strange color of the villa. Many references to the original colors have been provided. The case seems to end on April 14, 2023, when the People's Committee of Hoan Kiem District claimed that the current colors are just studies, not official color of the work. This article does not specifically discuss this case. The writer wants to take this opportunity to provide some backgr
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Heinanen, J., and R. Guerin. A Two Rate Three Color Marker. RFC Editor, 1999. http://dx.doi.org/10.17487/rfc2698.

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Chou, Stephen, and Daniel Tsui. Voltage Tunable Two-Color Quantum Well Infrared Photodetector. Defense Technical Information Center, 2003. http://dx.doi.org/10.21236/ada429217.

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St Pere, Robin R. A Correlation Study between Two Color-Measuring Spectrophotometers. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada231416.

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Jobes, F. C. Rational choices for the wavelengths of a two color interferometer. Office of Scientific and Technical Information (OSTI), 1995. http://dx.doi.org/10.2172/93727.

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Rohlfing, E. A., J. D. Tobiason, J. R. Dunlop, and S. Williams. Two-color resonant four-wave mixing: A tool for double resonance spectroscopy. Office of Scientific and Technical Information (OSTI), 1995. http://dx.doi.org/10.2172/106509.

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Currie, Douglas G., and Dennis D. Wellnitz. On Two Color and CCD Methods for the Determination of Astronomic Position. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada174088.

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Little, J. W., S. W. Kennedy, R. P. Leavitt, M. L. Lucas, and K. A. Olver. A New Two-Color Infrared Photodetector Design Using INGAAS/INALAS Coupled Quantum Wells. Defense Technical Information Center, 1999. http://dx.doi.org/10.21236/ada393876.

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Zholents, A. A. A proposal for a generation of two-color ultra-short x-ray pulses. Office of Scientific and Technical Information (OSTI), 2012. http://dx.doi.org/10.2172/1067990.

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Allen, S. J. Potential for Terahertz/Optical, Two Color Non-linear Sensing of Liquid Biochemical Agents. Defense Technical Information Center, 2011. http://dx.doi.org/10.21236/ada545156.

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