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The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j). -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
Building a Larger Class of Graphs for Efficient Reconfiguration
Building a larger class of graphs for efficient reconfiguration of vertex colouring by Owen Merkel A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2020 c Owen Merkel 2020 This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Statement of Contributions Owen Merkel is the sole author of Chapters 1, 2, 3, and 6 which were written under the supervision of Prof. Therese Biedl and Prof. Anna Lubiw and were not written for publication. Exceptions to sole authorship are Chapters 4 and 5, which is joint work with Prof. Therese Biedl and Prof. Anna Lubiw and consists in part of a manuscript written for publication. iii Abstract A k-colouring of a graph G is an assignment of at most k colours to the vertices of G so that adjacent vertices are assigned different colours. The reconfiguration graph of the k-colourings, Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge in Rk(G) if they differ in colour on exactly one vertex. For a k-colourable graph G, we investigate the connectivity and diameter of Rk+1(G). It is known that not all weakly chordal graphs have the property that Rk+1(G) is connected. -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first. -
Blue-Red Hackenbush
Basic rules Two players: Blue and Red. Perfect information. Players move alternately. First player unable to move loses. The game must terminate. Mathematical Games – p. 1 Outcomes (assuming perfect play) Blue wins (whoever moves first): G > 0 Red wins (whoever moves first): G < 0 Mover loses: G = 0 Mover wins: G0 Mathematical Games – p. 2 Two elegant classes of games number game: always disadvantageous to move (so never G0) impartial game: same moves always available to each player Mathematical Games – p. 3 Blue-Red Hackenbush ground prototypical number game: Blue-Red Hackenbush: A player removes one edge of his or her color. Any edges not connected to the ground are also removed. First person unable to move loses. Mathematical Games – p. 4 An example Mathematical Games – p. 5 A Hackenbush sum Let G be a Blue-Red Hackenbush position (or any game). Recall: Blue wins: G > 0 Red wins: G < 0 Mover loses: G = 0 G H G + H Mathematical Games – p. 6 A Hackenbush value value (to Blue): 3 1 3 −2 −3 sum: 2 (Blue is two moves ahead), G> 0 3 2 −2 −2 −1 sum: 0 (mover loses), G= 0 Mathematical Games – p. 7 1/2 value = ? G clearly >0: Blue wins mover loses! x + x - 1 = 0, so x = 1/2 Blue is 1/2 move ahead in G. Mathematical Games – p. 8 Another position What about ? Mathematical Games – p. 9 Another position What about ? Clearly G< 0. Mathematical Games – p. 9 −13/8 8x + 13 = 0 (mover loses!) x = -13/8 Mathematical Games – p. -
Combinatorial Game Theory
Combinatorial Game Theory Aaron N. Siegel Graduate Studies MR1EXLIQEXMGW Volume 146 %QIVMGER1EXLIQEXMGEP7SGMIX] Combinatorial Game Theory https://doi.org/10.1090//gsm/146 Combinatorial Game Theory Aaron N. Siegel Graduate Studies in Mathematics Volume 146 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 91A46. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-146 Library of Congress Cataloging-in-Publication Data Siegel, Aaron N., 1977– Combinatorial game theory / Aaron N. Siegel. pages cm. — (Graduate studies in mathematics ; volume 146) Includes bibliographical references and index. ISBN 978-0-8218-5190-6 (alk. paper) 1. Game theory. 2. Combinatorial analysis. I. Title. QA269.S5735 2013 519.3—dc23 2012043675 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
Results on the Grundy Chromatic Number of Graphs Manouchehr Zaker
Discrete Mathematics 306 (2006) 3166–3173 www.elsevier.com/locate/disc Results on the Grundy chromatic number of graphs Manouchehr Zaker Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran Received 7 November 2003; received in revised form 22 June 2005; accepted 22 June 2005 Available online 7 August 2006 Abstract Given a graph G,byaGrundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i<j, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by (G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining (G)k, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus–Gaddum-type inequalities concerning (G) and (Gc), for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with (G) colors. Equivalently G is well colored if (G) = (G). We prove that the recognition problem of well-colored graphs is a coNP-complete problem. © 2006 Elsevier B.V. All rights reserved. Keywords: Colorings; Chromatic number; Grundy number; First-fit colorings; NP-complete; Edge dominating sets 1. -
Math Book from Wikipedia
Math book From Wikipedia PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Mon, 25 Jul 2011 10:39:12 UTC Contents Articles 0.999... 1 1 (number) 20 Portal:Mathematics 24 Signed zero 29 Integer 32 Real number 36 References Article Sources and Contributors 44 Image Sources, Licenses and Contributors 46 Article Licenses License 48 0.999... 1 0.999... In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the only representation. The non-terminating form is more convenient for understanding the decimal expansions of certain fractions and, in base three, for the structure of the ternary Cantor set, a simple fractal. -
Three Years of Graphs and Music: Some Results in Graph Theory and Its
Three years of graphs and music : some results in graph theory and its applications Nathann Cohen To cite this version: Nathann Cohen. Three years of graphs and music : some results in graph theory and its applications. Discrete Mathematics [cs.DM]. Université Nice Sophia Antipolis, 2011. English. tel-00645151 HAL Id: tel-00645151 https://tel.archives-ouvertes.fr/tel-00645151 Submitted on 26 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ DE NICE-SOPHIA ANTIPOLIS - UFR SCIENCES ECOLE´ DOCTORALE STIC SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION TH ESE` pour obtenir le titre de Docteur en Sciences de l’Universite´ de Nice - Sophia Antipolis Mention : INFORMATIQUE Present´ ee´ et soutenue par Nathann COHEN Three years of graphs and music Some results in graph theory and its applications These` dirigee´ par Fred´ eric´ HAVET prepar´ ee´ dans le Projet MASCOTTE, I3S (CNRS/UNS)-INRIA soutenue le 20 octobre 2011 Rapporteurs : Jørgen BANG-JENSEN - Professor University of Southern Denmark, Odense Daniel KRAL´ ’ - Associate -
Combinatorial Game Theory: an Introduction to Tree Topplers
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2015 Combinatorial Game Theory: An Introduction to Tree Topplers John S. Ryals Jr. Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Part of the Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Ryals, John S. Jr., "Combinatorial Game Theory: An Introduction to Tree Topplers" (2015). Electronic Theses and Dissertations. 1331. https://digitalcommons.georgiasouthern.edu/etd/1331 This thesis (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. (Under the Direction of Hua Wang) ABSTRACT The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing. Key Words: combinatorial game theory, tree topplers, domineering, hackenbush 2009 Mathematics Subject Classification: 91A46 COMBINATORIAL GAME THEORY: AN INTRODUCTION TO TREE TOPPLERS by JOHN S. RYALS, JR. B.S. in Applied Mathematics A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 2015 c 2015 JOHN S. -
The Edge Grundy Numbers of Some Graphs
b b International Journal of Mathematics and M Computer Science, 12(2017), no. 1, 13–26 CS The Edge Grundy Numbers of Some Graphs Loren Anderson1, Matthew DeVilbiss2, Sarah Holliday3, Peter Johnson4, Anna Kite5, Ryan Matzke6, Jessica McDonald4 1School of Mathematics University of Minnesota Twin Cities Minneapolis, MN 55455, USA 2Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045, USA 3Department of Mathematics Kennesaw State University Kennesaw, GA 30144, USA 4Department of Mathematics and Statistics Auburn University Auburn, AL 36849, USA 5Department of Mathematics Wayland Baptist University Plainview, TX 79072, USA 6Department of Mathematics Gettysburg College Gettysburg, PA 17325, USA email: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] (Received October 27, 2016, Accepted November 19, 2016) Key words and phrases: Grundy coloring, edge Grundy coloring, line graph, grid. AMS (MOS) Subject Classifications: 05C15. This research was supported by NSF grant no. 1262930. ISSN 1814-0432, 2017 http://ijmcs.future-in-tech.net 14 Anderson, DeVilbiss, Holliday, Johnson, Kite, Matzke, McDonald Abstract The edge Grundy numbers of graphs in a number of different classes are determined, notably for the complete and the complete bipartite graphs, as well as for the Petersen graph, the grids, and the cubes. Except for small-graph exceptions, these edge Grundy numbers turn out to equal a natural upper bound of the edge Grundy number, or to be one less than that bound. 1 Grundy colorings and Grundy numbers A Grundy coloring of a (finite simple) graph G is a proper coloring of the vertices of G with positive integers with the property that if v ∈ V (G) is colored with c > 1, then all the colors 1,...,c − 1 appear on neighbors of v. -
Lower Bounds for the Partial Grundy Number of the Lexicographic Product of Graphs
Lower Bounds for the Partial Grundy Number of the Lexicographic Product of Graphs Kenny Domingues , Yuri Silva de Oliveira , Ana Silva 1ParGO Group - Parallelism, Graphs and Optimization Centro de Ciências - Departamento de Matemática Universidade Federal do Ceará (UFC) fkennywille,[email protected], [email protected] Abstract. A Grundy coloring of a graph G is a coloring obtained by applying the greedy algorithm according to some order of the vertices of G. The Grundy number of G is then the largest k such that G has a greedy coloring with k colors. A partial Grundy coloring is a coloring where each color class contains at least one greedily colored vertex, and the partial Grundy number of G is the largest k for which G has a partial greedy coloring. In this article, we give some results on the partial Grundy number of the lexicographic product of graphs, drawing a parallel with known results for the Grundy number. 1. Introduction The definitions and notations used in this paper are standard and can be found in any graph theory book. Also, we consider only simple graphs. Graph coloring problems are perhaps the most studied problems in graph theory, due to their practical and theoretical importance. A independent set in a graph G is a subset S ⊆ V (G) such that uv2 = E(G) for every u; v 2 S.A proper k-coloring (from now on called simply a coloring) of a graph G is a partition of V (G) into k independent sets, and the chromatic number of G is the minimum value χ(G) for which G has a proper k-coloring.