DOCSLIB.ORG

Independence Number of Graphs A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Independence Number of Graphs A Discrete Mathematics xxx (xxxx) xxx Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On the k-independence number of graphs A. Abiad a,b, G. Coutinho c, M.A. Fiol d a Department of Quantitative Economics, Maastricht University, Maastricht, The Netherlands b Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium c Department of Computer Science, Federal University of Minas Gerais, Belo Horizonte, Brazil d Departament de Matemàtiques, Barcelona Graduate School of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain article info a b s t r a c t Article history: This paper generalizes and unifies the existing spectral bounds on the k-independence Received 19 March 2018 number of a graph, which is the maximum size of a set of vertices at pairwise distance Accepted 18 January 2019 greater than k. The previous bounds known in the literature follow as a corollary of the Available online xxxx main results in this work. We show that for most cases our bounds outperform the previous Keywords: known bounds. Some infinite families of graphs where the bounds are tight are also Graph presented. Finally, as a byproduct, we derive some spectral lower bounds for the diameter k-independence number of a graph. Spectrum ' 2019 Elsevier B.V. All rights reserved. Interlacing Regular partition Antipodal distance-regular graph 1. Introduction Given a graph G, let αk D αk(G) denote the size of the largest set of vertices such that any two vertices in the set are at distance larger than k. This choice of notation is no coincidence, since actually α1 is just the independence number of a graph. The parameter αk(G) therefore represents the largest number of vertices which can be k C 1 spread out in G. It is known that determining αk is NP-Hard in general [19]. The k-independence number of a graph is directly related to other combinatorial parameters such as the average distance [13], packing chromatic number [14], injective chromatic number [17], and strong chromatic index [21]. Upper bounds on the k-independence number directly give lower bounds on the corresponding distance or packing chromatic number. Regarding it, Alon and Mohar [2] asked for the extremal value of the distance chromatic number for graphs of a given girth and degree. In this paper we generalize and improve the known spectral upper bounds for the k-independence number from [1,8]. For some cases, we also show that our bounds are sharp. As far as we are aware, there seems to be some conflict in the existing literature regarding the use of the term `k-independence number'. The following list contains the three conflicting definitions, which all, nonetheless, are a natural generalization of the concept of independence number. 1. Caro and Hansberg [6] use the term `k-independence number' to denote the maximum size of a set of vertices in a graph whose induced subgraph has maximum degree k. Thus, α0 is the usual independence number. 2. Špacapan [22] uses `k-independence number' to denote the size of the largest k-colorable subgraph of G. With this notation, α1 stands for the usual independence number of G. E-mail addresses: [email protected] (A. Abiad), [email protected] (G. Coutinho), [email protected] (M.A. Fiol). https://doi.org/10.1016/j.disc.2019.01.016 0012-365X/' 2019 Elsevier B.V. All rights reserved. Please cite this article as: A. Abiad, G. Coutinho and M.A. Fiol, On the k-independence number of graphs, Discrete Mathematics (2019), https://doi.org/10.1016/j.disc.2019.01.016. 2 A. Abiad, G. Coutinho and M.A. Fiol / Discrete Mathematics xxx (xxxx) xxx 3. Fiol [8] and Abiad, Tait, and Cioabă [1] use `k-independence number' to denote the size of the largest set of vertices such that any two vertices in the set are at distance larger than k. The latter definition is the one we use in this work. The first known spectral bound for the independence number α is due to Cvetkovi¢ [7]. Theorem 1.1 (Cvetkovi¢ [7]). Let G be a graph with eigenvalues λ1 ≥ · · · ≥ λn. Then, α ≤ minfjfi V λi ≥ 0gj and jfi V λi ≤ 0gjg: Another well-known result is the following bound due to Hoffman (unpublished; see for instance Haemers [16]). Theorem 1.2 (Hoffman [16]). If G is a regular graph on n vertices with eigenvalues λ1 ≥ · · · ≥ λn, then −λn α ≤ n : λ1 − λn Regarding the k-independence number, the following three results are known. The first is due to Fiol [8] and requires a preliminary definition. Let G be a graph with distinct eigenvalues θ0 > ··· > θd. Let Pk(x) be chosen among all polynomials p(x) 2 Rk(x), that is, polynomials of real coefficients and degree at most k, satisfying jp(θi)j ≤ 1 for all i D 1;:::; d, and such that Pk(θ0) is maximized. The polynomial Pk(x) defined above is called the k-alternating polynomial of G and was shown to be unique in [12], where it was used to study the relationship between the spectrum of a graph and its diameter. Theorem 1.3 (Fiol [8]). Let G be a d-regular graph on n vertices, with distinct eigenvalues θ0 > ··· > θd and let Pk(x) be its k-alternating polynomial. Then, 2n αk ≤ : Pk(θ0) C 1 The second and third bounds are due to Abiad, Cioabă, and Tait [1]. The first is a Cvetkovi¢-like approach, whereas the second resembles Hoffman's. Theorem 1.4 (Abiad, Cioabă, Tait [1]). Let G be a graph on n vertices with adjacency matrix A, with eigenvalues λ1 ≥ · · · ≥ λn. k Let wk and Wk be respectively the smallest and the largest diagonal entries of A . Then, α λk w α λk : k ≤ jfi V i ≥ k(G)gj and k ≤ jfi V i ≤ Wk(G)gj Theorem 1.5 (Abiad, Cioabă, Tait [1]). Let G be a δ-regular graph on n vertices with adjacency matrix A, whose distinct eigenvalues θ δ > > θ 2 k θ θ ; θ are 0(D ) ··· d. Let Wfk be the largest diagonal entry of A C A C···C A . Let D maxfj 1j j djg. Then, C Pk θ j Wfk jD1 αk ≤ n : Pk δj C Pk θ j jD1 jD1 2. Preliminaries For basic notation and results see [3,15]. Let G D (V ; E) be a graph with n D jV j vertices, m D jEj edges, and adjacency D fθ > θ (m1) > ··· > θ (md)g matrix A with spectrum sp G 0 1 d , where the superscripts denote multiplicities. When the eigenvalues are presented with possible repetitions, we shall indicate them by λ1 ≥ λ2 ≥ · · · ≥ λn. Let us consider the scalar product in RdTxU: d 1 1 X hf ; giG D tr(f (A)g(A)) D mif (θi)g(θi): n n iD0 The so-called predistance polynomials p0(D 1); p1;:::; pd are a sequence of orthogonal polynomials with respect to the D k k2 D θ above product, with dgr pi i, and are normalized in such a way that pi G pi( 0) (this makes sense since it is known that pi(θ0) > 0) for i D 0;:::; d. Therefore they are uniquely determined, for instance, following the Gram–Schmidt process. They were introduced by Fiol and Garriga in [10] to prove the so-called `spectral excess theorem' for distance-regular graphs. We also use the sum polynomials qi D p0 C· · ·Cpi, for i D 0;:::; d−1, which are also a sequence of orthogonal polynomials, now with respect to the scalar product d−1 1 X hf ; giTGU D mi(θ0 − θi)f (θi)g(θi); n iD0 (see [11, Prop. 4.3] for a proof) and satisfy 1 D q0(θ0) < q1(θ0) < ··· < qd−1(θ0) < n. See [5] for further details and applications. Please cite this article as: A. Abiad, G. Coutinho and M.A. Fiol, On the k-independence number of graphs, Discrete Mathematics (2019), https://doi.org/10.1016/j.disc.2019.01.016. A. Abiad, G. Coutinho and M.A. Fiol / Discrete Mathematics xxx (xxxx) xxx 3 Eigenvalue interlacing is a powerful and old technique that has found countless applications in combinatorics and other fields. This technique will be used in several of our proofs. For more details, historical remarks and other applications see Fiol and Haemers [9,16]. Given square matrices A and B with respective eigenvalues λ1 ≥ · · · ≥ λn and µ1 ≥ · · · ≥ µm, with m < n, we say that the second sequence interlaces the first if, for all i D 1;:::; m, it follows that λi ≥ µi ≥ λn−mCi: Theorem 2.1 (Interlacing [9,16]). Let S be a real n × m matrix such that S T S D I, and let A be a n × n matrix with eigenvalues T λ1 ≥ · · · ≥ λn. Define B D S AS, and call its eigenvalues µ1 ≥ · · · ≥ µm. Then, (i) The eigenvalues of B interlace those of A. (ii) If µi D λi or µi D λn−mCi, then there is an eigenvector v of B for µi such that Sv is eigenvector of A for µi. (iii) If there is an integer k 2 f0;:::; mg such that λi D µi for 1 ≤ i ≤ k, and µi D λn−mCi for k C 1 ≤ i ≤ m (tight interlacing), then SB D AS. Two interesting particular cases where interlacing occurs (obtained by choosing appropriately the matrix S) are the following. Let A be the adjacency matrix of a graph G D (V ; E).
Load more

Recommended publications
  • Configurations of Points and Lines
    Configurations of Points and Lines "RANKO'RüNBAUM 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/103 Configurations of Points and Lines Configurations of Points and Lines Branko Grünbaum Graduate Studies in Mathematics Volume 103 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 05–03, 05B30, 05C62, 51–03, 51A20, 51A45, 51E30, 52C30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-103 Library of Congress Cataloging-in-Publication Data Gr¨unbaum, Branko. Configurations of points and lines / Branko Gr¨unbaum. p. cm. — (Graduate studies in mathematics ; v. 103) Includes bibliographical references and index. ISBN 978-0-8218-4308-6 (alk. paper) 1. Configurations. I. Title. QA607.G875 2009 516.15—dc22 2009000303 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 2009 by the American Mathematical Society.
    [Show full text]
  • Drawing Graphs and Maps with Curves
    Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts.
    [Show full text]
  • Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
    SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1.
    [Show full text]
  • Graph Theory with Applications
    GRAPH THEORY WITH APPLICATIONS J. A. Bondy and U. S. R. Murty Department of Combina tories and Optimization, University of Waterloo, Ontario, Canada NORfH-HOLLAND New York • Amsterdam • Oxford @J.A. Bondy and U.S.R. Muny 1976 First published in Great Britain 1976 by The Macmillan Press Ltd. First published in the U.S.A. 1976 by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, N.Y. 10017 Fifth Printing, 1982. Sole Distributor in the U.S.A: Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data Bondy, John Adrian. Graph theory with applications. Bibliography: p. lncludes index. 1. Graph theory. 1. Murty, U.S.R., joint author. II. Title. QA166.B67 1979 511 '.5 75-29826 ISBN 0.:444-19451-7 AU rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Printed in the United States of America To our parents Preface This book is intended as an introduction to graph theory. Our aim bas been to present what we consider to be the basic material, together with a wide variety of applications, both to other branches of mathematics and to real-world problems. Included are simple new proofs of theorems of Brooks, Chvâtal, Tutte and Vizing. The applications have been carefully selected, and are treated in some depth. We have chosen to omit ail so-called 'applications' that employ just the language of graphs and no theory. The applications appearing at the end of each chapter actually make use of theory developed earlier in the same chapter.
    [Show full text]
  • The Intriguing Human Preference for a Ternary Patterned Reality
    75 Kragujevac J. Sci. 27 (2005) 75-114. THE INTRIGUING HUMAN PREFERENCE FOR A TERNARY PATTERNED REALITY Lionello Pogliani,* Douglas J. Klein,‡ and Alexandru T. Balaban¥ *Dipartimento di Chimica Università della Calabria, 87030 Rende (CS), Italy; ‡Texas A&M University at Galveston, Galveston, Texas, 77553-1675, USA; ¥ Texas A&M University at Galveston, Galveston, Texas, 77551, USA E-mails: [email protected]; [email protected]; balabana @tamug.tamu.edu (Received February 10, 2005) Three paths through the high spring grass / One is the quicker / and I take it (Haikku by Yosa Buson, Spring wind on the river bank Kema) How did number theory degrade to numerology? How did numerology influence different aspects of human creation? How could some numbers assume such an extraordinary meaning? Why some numbers appear so often throughout the fabric of reality? Has reality really a preference for a ternary pattern? Present excursion through the ‘life’ and ‘deeds’ of this pattern and of its parent number in different fields of human culture try to offer an answer to some of these questions giving at the same time a glance of the strange preference of the human mind for a patterned reality based on number three, which, like the 'unary' and 'dualistic' patterned reality, but in a more emphatic way, ended up assuming an archetypical character in the intellectual sphere of humanity. Introduction Three is not only the sole number that is the sum of all the natural numbers less than itself, but it lies between arguably the two most important irrational and transcendental numbers of all mathematics, i.e., e < 3 < π.
    [Show full text]
  • XYZ Decompositions of Graphs Dagstuhl Seminar 08191
    XYZ decom- positions V. Batagelj XYZ decompositions of graphs Dagstuhl Seminar 08191 Motivation XYZ decom- Vladimir Batagelj positions Results University of Ljubljana, FMF, Dept. of Mathematics; and References IMFM Ljubljana, Dept. of Theoretical Computer Science joint work with Franz J. Brandenburg, Walter Didimo, Giuseppe Liotta, Maurizio Patrignani 23rd Leoben-Ljubljana Graph Theory Seminar Ljubljana, 14-15. November 2008 V. Batagelj XYZ decompositions Outline XYZ decom- positions V. Batagelj Dagstuhl Seminar 08191 Motivation 1 XYZ decom- Dagstuhl Seminar 08191 positions 2 Motivation Results 3 XYZ decompositions References 4 Results 5 References V. Batagelj XYZ decompositions Dagstuhl Seminar 08191 XYZ decom- positions V. Batagelj Dagstuhl Seminar 08191 Motivation XYZ decom- positions Results References Dagstuhl Seminar 08191: Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl, May 4-9, 2008 V. Batagelj XYZ decompositions Motivation XYZ decom- positions For dense graphs the matrix representation is better [3]. Pajek - shadow [0.00,1.00] V. Batagelj uki fra wge jap net ita usa bel lux car swe kod den swi can nor par spa irn Dagstuhl yem mat irq pak ire nau aut bol hun Seminar 08191 lib hai isr nic sau alb kuw aus gua mon fin por zai hon bra arg pol syr cze kmr usr Motivation tri ege yug sri ind cha saf gre tur sie egy bul rum uru tai rum syr XYZ decom- leb upv cyp vnr ice arg tun gab cub liy positions pol mor usr lux alg spanor nig isr mli uga ins bul liy ken mex usa eth brm pan net tha cze fra jor sud gre
    [Show full text]
  • On Finding Hajós Constructions Erik Johnson
    On Finding Haj´osConstructions by Erik Johnson A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Computing Science University of Alberta c Erik Johnson, 2015 Abstract We develop a method for searching for Haj´osconstructions. Our results include the discovery of new constructions for some well-known graphs, including the Gr¨otzsch graph, Chv´atalgraph, and Brinkmann graph; also, we prove that the first two of these are shortest possible constructions. These are the first published constructions for these graphs. ii Acknowledgments Thank you Dr. Joseph Culberson for getting me started on this topic and for your ongoing help and interest. Thank you Dr. Ryan Hayward for seeing this project through and for all of your help in advising me on my thesis. I would also like to thank Jeanette Bliemel for her patience, support, and encouragement as I took my meandering path through work and academics. iii Table of Contents 1 Introduction 1 1.1 Graph Colouring and Complexity Theory ............ 2 1.2 Contributions ........................... 7 2 The Gr¨otzsch Graph 8 2.1 Background ............................ 8 2.2 Construction ........................... 9 2.3 Minimality of the Construction ................. 9 2.4 Alternate Constructions of the Gr¨otzsch Graph . 16 3 Variations on Haj´osConstructions 18 3.1 Tree-Like Haj´osConstructions . 18 3.2 Haj´os'Proof of Completeness . 22 3.3 Top-Down Constructions ..................... 23 4 Searching for Haj´osConstructions 27 4.1 Background ............................ 27 4.2 Haj´os'Algorithm ......................... 28 4.3 Searching the Constructions Space . 29 4.4 Isomorphisms and Tree-Like Constructions .
    [Show full text]
  • Vertex-Transitive Graphs That Have No Hamilton Decomposition
    Vertex-transitive graphs that have no Hamilton decomposition Darryn Bryant ∗ Matthew Dean † Abstract It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency. 1 Introduction A famous question of Lov´asz concerns the existence of Hamilton paths in vertex-transitive graphs [28], and no example of a connected vertex-transitive graph with no Hamilton path is known. The related question concerning the existence of Hamilton cycles in vertex-transitive graphs is another interesting and well-studied problem in graph theory, see the survey [23]. A Hamiltonian graph is a graph containing a Hamilton cycle. Thomassen (see [10, 23]) has conjectured that there are only finitely many non-Hamiltonian connected vertex-transitive graphs. On the other hand, Babai [8, 9] has conjectured that there are infinitely many such graphs. To date only five are known. These arXiv:1408.5211v3 [math.CO] 12 Nov 2014 are the complete graph of order 2, the Petersen graph, the Coxeter graph, and the two graphs obtained from the Petersen and Coxeter graphs by replacing each vertex with a triangle. For a regular graph of valency at least 4, a stronger property than the existence of a Hamilton cycle is the existence of a Hamilton decomposition. If X is a k-valent graph, then a Hamilton k decomposition of X is a set of ⌊ 2 ⌋ pairwise edge-disjoint Hamilton cycles in X. Given the small number of non-Hamiltonian connected vertex-transitive graphs, and the uncertainty concerning the existence of others, it is natural to ask how many connected vertex-transitive graphs have no Hamilton decomposition.
    [Show full text]
  • Hypercubes Are Determined by Their Distance Spectra 3
    HYPERCUBES ARE DETERMINED BY THEIR DISTANCE SPECTRA JACK H. KOOLEN♦, SAKANDER HAYAT♠, AND QUAID IQBAL♣ Abstract. We show that the d-cube is determined by the spectrum of its distance matrix. 1. Introduction For undefined terms, see next section. For integers n 2 and d 2, the Ham- ming graph H(d, n) has as vertex set, the d-tuples with elements≥ from≥0, 1, 2,...,n 1 , with two d-tuples are adjacent if and only if they differ only in one{ coordinate.− For} a positive integer d, the d-cube is the graph H(d, 2). In this paper, we show that the d-cube is determined by its distance spectrum, see Theorem 5.4. Observe that the d-cube has exactly three distinct distance eigenvalues. Note that the d-cube is not determined by its adjacency spectrum as the Hoffman graph [11] has the same adjacency spectrum as the 4-cube and hence the cartesian product of (d 4)-cube and the Hoffman graph has the same adjacency spectrum − as the d-cube (for d 4, where the 0-cube is just K1). This also implies that the complement of the d-cube≥ is not determined by its distance spectrum, if d 4. ≥ There are quite a few papers on matrices of graphs with a few distinct eigenvalues. An important case is, when the Seidel matrix of a graph has exactly two distinct eigenvalues is very much studied, see, for example [18]. Van Dam and Haemers [20] characterized the graphs whose Laplacian matrix has two distinct nonzero eigenval- ues.
    [Show full text]
  • A Survey of Graph Coloring - Its Types, Methods and Applications
    FOUNDATIONS OF COMPUTING AND DECISION SCIENCES Vol. 37 (2012) No. 3 DOI: 10.2478/v10209-011-0012-y A SURVEY OF GRAPH COLORING - ITS TYPES, METHODS AND APPLICATIONS Piotr FORMANOWICZ1;2, Krzysztof TANA1 Abstract. Graph coloring is one of the best known, popular and extensively researched subject in the eld of graph theory, having many applications and con- jectures, which are still open and studied by various mathematicians and computer scientists along the world. In this paper we present a survey of graph coloring as an important subeld of graph theory, describing various methods of the coloring, and a list of problems and conjectures associated with them. Lastly, we turn our attention to cubic graphs, a class of graphs, which has been found to be very interesting to study and color. A brief review of graph coloring methods (in Polish) was given by Kubale in [32] and a more detailed one in a book by the same author. We extend this review and explore the eld of graph coloring further, describing various results obtained by other authors and show some interesting applications of this eld of graph theory. Keywords: graph coloring, vertex coloring, edge coloring, complexity, algorithms 1 Introduction Graph coloring is one of the most important, well-known and studied subelds of graph theory. An evidence of this can be found in various papers and books, in which the coloring is studied, and the problems and conjectures associated with this eld of research are being described and solved. Good examples of such works are [27] and [28]. In the following sections of this paper, we describe a brief history of graph coloring and give a tour through types of coloring, problems and conjectures associated with them, and applications.
    [Show full text]
  • Semisymmetric Graphs from Polytopes
    Semisymmetric Graphs from Polytopes Barry Monson¤ University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 ([email protected]) Toma·z Pisanskiy University of Ljubljana, FMF Jadranska 19, Ljubljana 1111, Slovenia; and Faculty of Education, University of Primorska, Cankarjeva 5, Koper 6000, Slovenia ([email protected]) Egon Schultez Northeastern University Boston MA 02115, USA ([email protected]) Asia Ivi¶c Weissx York University Toronto, Ontario, Canada M3J 1P3 ([email protected]) Abstract Every ¯nite, self-dual, regular (or chiral) 4-polytope of type f3;q;3g has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by drop- ping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes. AMS Subject Classi¯cation (1991): Primary: 05C25. Secondary: 51M20. ¤Supported by the NSERC of Canada, grant 4818 ySupported by Ministry of Higher Education, Science and Technolgy of Slovenia grants P1-0294,J1-6062,L1-7230. zSupported by NSA-grant H98230-05-1-0027 xSupported by the NSERC of Canada, grant 8857 1 1 Introduction The theory of symmetric trivalent graphs and the theory of regular polytopes are each abundant sources of beautiful mathematical ideas. In [22], two of the authors established some general and unexpected connections between the two subjects, building upon a rich variety of examples appearing in the literature (see [4], [7], [10], [11], [28] and [29]). Here we develop these connections a little further, with speci¯c focus on semisymmetric graphs.
    [Show full text]
  • SAT Approach for Decomposition Methods
    SAT Approach for Decomposition Methods DISSERTATION submitted in partial fulfillment of the requirements for the degree of Doktorin der Technischen Wissenschaften by M.Sc. Neha Lodha Registration Number 01428755 to the Faculty of Informatics at the TU Wien Advisor: Prof. Stefan Szeider Second advisor: Prof. Armin Biere The dissertation has been reviewed by: Daniel Le Berre Marijn J. H. Heule Vienna, 31st October, 2018 Neha Lodha Technische Universität Wien A-1040 Wien Karlsplatz 13 Tel. +43-1-58801-0 www.tuwien.ac.at Erklärung zur Verfassung der Arbeit M.Sc. Neha Lodha Favoritenstrasse 9, 1040 Wien Hiermit erkläre ich, dass ich diese Arbeit selbständig verfasst habe, dass ich die verwen- deten Quellen und Hilfsmittel vollständig angegeben habe und dass ich die Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken oder dem Internet im Wortlaut oder dem Sinn nach entnommen sind, auf jeden Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht habe. Wien, 31. Oktober 2018 Neha Lodha iii Acknowledgements First and foremost, I would like to thank my advisor Prof. Stefan Szeider for guiding me through my journey as a Ph.D. student with his patience, constant motivation, and immense knowledge. His guidance helped me during the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor. Next, I would like to thank my co-advisor Prof. Armin Biere, my reviewers, Prof. Daniel Le Berre and Dr. Marijn Heule, and my Ph.D. committee, Prof. Reinhard Pichler and Prof. Florian Zuleger, for their insightful comments, encouragement, and patience.
    [Show full text]