Complexity Results for Rainbow Matchings
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Linear K-Arboricities on Trees
Discrete Applied Mathematics 103 (2000) 281–287 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Note Linear k-arboricities on trees Gerard J. Changa; 1, Bor-Liang Chenb, Hung-Lin Fua, Kuo-Ching Huangc; ∗;2 aDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Business Administration, National Taichung Institue of Commerce, Taichung 404, Taiwan cDepartment of Applied Mathematics, Providence University, Shalu 433, Taichung, Taiwan Received 31 October 1997; revised 15 November 1999; accepted 22 November 1999 Abstract For a ÿxed positive integer k, the linear k-arboricity lak (G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algo- rithm to determine whether a tree T has lak (T)6m. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Linear forest; Linear k-forest; Linear arboricity; Linear k-arboricity; Tree; Leaf; Penultimate vertex; Algorithm; NP-complete 1. Introduction All graphs in this paper are simple, i.e., ÿnite, undirected, loopless, and without multiple edges. A linear k-forest is a graph whose components are paths of length at most k.Alinear k-forest partition of G is a partition of the edge set E(G) into linear k-forests. The linear k-arboricity of G, denoted by lak (G), is the minimum size of a linear k-forest partition of G. -
Coloring Problems in Graph Theory Kacy Messerschmidt Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2018 Coloring problems in graph theory Kacy Messerschmidt Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Messerschmidt, Kacy, "Coloring problems in graph theory" (2018). Graduate Theses and Dissertations. 16639. https://lib.dr.iastate.edu/etd/16639 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Coloring problems in graph theory by Kacy Messerschmidt A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Bernard Lidick´y,Major Professor Steve Butler Ryan Martin James Rossmanith Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2018 Copyright c Kacy Messerschmidt, 2018. All rights reserved. TABLE OF CONTENTS LIST OF FIGURES iv ACKNOWLEDGEMENTS vi ABSTRACT vii 1. INTRODUCTION1 2. DEFINITIONS3 2.1 Basics . .3 2.2 Graph theory . .3 2.3 Graph coloring . .5 2.3.1 Packing coloring . .6 2.3.2 Improper coloring . -
Structural Parameterizations of Clique Coloring
Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e. -
Rainbow Matchings and Transversals∗
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 59(1) (2014), Pages 211–217 Rainbow matchings and transversals∗ Janos´ Barat´ † MTA-ELTE Geometric and Algebraic Combinatorics Research Group 1117 Budapest, P´azm´any P´eter s´et´any 1/C Hungary Ian M. Wanless School of Mathematical Sciences Monash University Clayton, Vic 3800 Australia Abstract We show that there exists a bipartite graph containing n matchings of 2 sizes mi n satisfying i mi = n + n/2−1, such that the matchings have no rainbow matching. This answers a question posed by Aharoni, Charbit and Howard. We also exhibit (n − 1) × n latin rectangles that cannot be decom- posed into transversals, and some related constructions. In the process we answer a question posed by H¨aggkvist and Johansson. Finally, we propose a Hall-type condition for the existence of a rain- bow matching. Two edges in a graph are independent if they do not share an endpoint. A matching is a set of edges that are pairwise independent. If M1,M2,...,Mn are matchings and there exist edges e1 ∈ M1, e2 ∈ M2,...,en ∈ Mn that form a matching then we say that {M1,M2,...,Mn} possesses a rainbow matching.Notethatwedo not require the matchings Mi to be disjoint, so it may even be the case that ej ∈ Mi for some j = i. A k × n matrix R, containing symbols from an alphabet Λ, can be viewed as listing edges in a bipartite graph GR. The two parts of GR correspond respectively to the columns of R, and to Λ. Each cell in R corresponds to an edge in GR that links the column and the symbol in the cell. -
Parameterized Algorithms and Kernels for Rainbow Matching
Parameterized Algorithms and Kernels for Rainbow Matching Sushmita Gupta1, Sanjukta Roy2, Saket Saurabh3, and Meirav Zehavi4 1 University of Bergen, Norway [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 3 University of Bergen, Norway, and The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 4 University of Bergen, Norway [email protected] Abstract In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow √ k ? 1+ 5 Matching on paths in time O ( 2 ) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O?(2k) and polynomial-space. Here, we show how a result by Björklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs. Keywords and phrases Rainbow Matching, Parameterized Algorithm, Bounded Search Trees, Divide-and-Conquer, 3-Set Packing, 3-Dimensional Matching Digital Object Identifier 10.4230/LIPIcs.MFCS.2017.71 1 Introduction The classical notion of matching has been extensively studied for several decades in the area of Combinatorial Optimization [6, 14]. -
Existences of Rainbow Matchings and Rainbow Matching Covers Lo, Allan
CORE Metadata, citation and similar papers at core.ac.uk Provided by University of Birmingham Research Portal University of Birmingham Existences of rainbow matchings and rainbow matching covers Lo, Allan DOI: 10.1016/j.disc.2015.05.015 License: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) Document Version Publisher's PDF, also known as Version of record Citation for published version (Harvard): Lo, A 2015, 'Existences of rainbow matchings and rainbow matching covers', Discrete Mathematics, vol. 338, pp. 2119-2124. https://doi.org/10.1016/j.disc.2015.05.015 Link to publication on Research at Birmingham portal Publisher Rights Statement: Eligibility for repository : checked 16/06/2015 General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. -
Choosability on H-Free Graphs?
Choosability on H-Free Graphs? Petr A. Golovach1, Pinar Heggernes1, Pim van 't Hof1, and Dani¨el Paulusma2 1 Department of Informatics, University of Bergen, Norway fpetr.golovach,pinar.heggernes,[email protected] 2 School of Engineering and Computing Sciences, Durham University, UK [email protected] Abstract. A graph is H-free if it has no induced subgraph isomorphic to H. We determine the computational complexity of the Choosability problem restricted to H-free graphs for every graph H that does not belong to fK1;3;P1 + P2;P1 + P3;P4g. We also show that if H is a linear forest, then the problem is fixed-parameter tractable when parameterized by k. Keywords. Choosability, H-free graphs, linear forest. 1 Introduction Graph coloring is without doubt one of the most fundamental concepts in both structural and algorithmic graph theory. The well-known Coloring problem, which takes as input a graph G and an integer k, asks whether G admits a k- coloring, i.e., a mapping c : V (G) ! f1; 2; : : : ; kg such that c(u) 6= c(v) whenever uv 2 E(G). The corresponding problem where k is fixed, i.e., not part of the input, is denoted by k-Coloring. Since graph coloring naturally arises in a vast number of theoretical and practical applications, it is not surprising that many variants and generalizations of the Coloring problem have been studied over the years. There are some very good surveys [1,28] and a book [22] on the subject. One well-known generalization of graph coloring, called List Coloring, was introduced by Vizing [29] and Erd}os,Rubin and Taylor [7]. -
Benny Sudakov Curriculum Vitae
BENNY SUDAKOV CURRICULUM VITAE Affiliation Professor, Department of Mathematics, ETH, Zurich, 8092, Switzerland. E-mail: [email protected]. Research interests Algebraic and Probabilistic Methods in Combinatorics, Graph Theory, Extremal Combinatorics, Ramsey Theory, Random Structures, Applications of Combinatorics to Theory of Computer Science. Academic appointments 2013- : Professor, Department of Mathematics, ETH, Z¨urich, Switzerland. 2015 (Fall): Fellow of Merton College, Oxford University. 2007-2014 : Professor, Department of Mathematics, UCLA, Los Angeles, USA. 2012 (Spring): Leverhulme Visiting Professor, Cambridge University and Fellow of Trinity College. 2002-2007 : Assistant Professor, Department of Mathematics, Princeton University, USA. 1999-2002 : Veblen Instructor, Department of Mathematics, Princeton University and Institute for Advanced Study, Princeton, USA. 2003 (Spring), 2005-2006, 2007 (Fall) : Member, Institute for Advanced Study, Princeton, USA. Education 1995-1999 : Ph. D. in Mathematics (with distinction), Tel Aviv University, Tel Aviv, Israel. 1991-1993: M. Sc. in Mathematics (summa cum laude), Tel Aviv University, Tel Aviv, Israel. 1990: B. Sc. in Mathematics (summa cum laude), Georgian State University, Tbilisi, USSR. Honors and Awards 2019: Member of the Academia Europaea. 2014: Humboldt Research Award, Germany. 2013: Fellow of the American Mathematical Society, Inaugural Class. 2010: Invited speaker, Combinatorics section, International Congress (ICM), Hyderabad, India. 2007-2011: David Saxon Presidential Term Chair in Mathematics, UCLA, USA. 2006-2011: NSF CAREER award. 2004-2006: Alfred P. Sloan Fellowship, USA. Grants 2013-2016, 2017-2020, 2020-2024: Swiss National Science Foundation grants. 1 2001-2004, 2004-2007, 2006-2011, 2011-2014: USA National Science Foundation grants. 2005-2009, 2009-2013: USA-Israel Binational Science Foundation grants. Ph. -
Linear Coloring of Graphs Raphael Yuster* Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel
DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 185 (1998) 293-297 Note Linear coloring of graphs Raphael Yuster* Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel Received 27 September 1996; received in revised form 10 July 1997; accepted 28 July 1997 Abstract A proper vertex coloring of a graph is called linear if the subgraph induced by the vertices colored by any two colors is a set of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by lc(G), is the minimum number of colors in a linear coloring of G. Extending a result of Alon, McDiarmid and Reed concerning acyclic graph colorings, we show that if G has maximum degree d then lc(G) = O(d3/2). We also construct explicit graphs with maximum degree d for which lc(G)= Q(d3/2), thus showing that the result is optimal, up to an absolute constant factor. (~) 1998 Elsevier Science B.V. All rights reserved I. Introduction All graphs considered here are finite, undirected and simple. For the standard graph- theoretic notations the reader is referred to [3]. A proper vertex coloring of a graph is called linear if the subgraph induced by the union of any two color classes is a linear forest, i.e. a set of vertex-disjoint paths. Let lc(G) be the minimum number of colors in a linear coloring of the graph G. We call lc(G) the linear chromatic number of G. A linear coloring is a special case of an acyclic coloring, where it is required that the subgraph induced by the union of any two color classes is a forest. -
On Latin Squares and Avoidable Arrays for Daniel, and for What Springtime Brings
On Latin squares and avoidable arrays For Daniel, and for what springtime brings Lina J. Andr´en: On Latin squares and avoidable arrays c 2010 Lina J. Andr´en Tryck: Print & Media, Ume˚auniversitet, Ume˚a isbn 978-91-7459-060-9 issn 1102-8300 Doctoral thesis No. 46, 2010, Department of Mathematics and Mathematical statistics, Ume˚auniversity On Latin squares and avoidable arrays Lina J. Andr´en Doctoral thesis no. 46 Department of Mathematics and Mathematical statistics Ume˚aUniversity, 2010 Contents 1 Summary of papers 1 2 Introduction 3 3 Basic definitions and tools 5 3.1 Latin squares and arrays . 5 3.2 Graphs and colorings . 10 4 Problems and known results 13 5 Structural properties of Latin squares 21 5.1 Cycles and homogeneity . 21 5.2 Transversals and rainbow matchings . 22 6 Research on avoidability and list-edge colorings 25 6.1 Someusefultechniques. 25 Bibliography 33 v Abstract This thesis consists of the four papers listed below and a survey of the research area. I Lina J. Andr´en: Avoiding (m, m, m)-arrays of order n =2k II Lina J. Andr´en: Avoidability of random arrays III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order IV Lina J. Andr´en, Carl Johan Casselgren and Lars-Daniel Ohman:¨ Avoiding arrays of odd order by Latin squares Papers I, III and IV are all concerned with a conjecture by H¨aggkvist saying that there is a constant c such that for any positive integer n,ifm cn, then for every n n array A of subsets of 1,...,n such that no cell contains≤ a set of size greater× than m, and none of the{ elements} 1,...,nbelongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1,...,n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. -
Short Tours Through Large Linear Forests
Improved Approximations for Graph-TSP in Regular Graphs R Ravi Carnegie Mellon University Joint work with Uriel Feige (Weizmann), Satoru Iwata (U Tokyo), Jeremy Karp (CMU), Alantha Newman (G-SCOP) and Mohit Singh (MSR) 1 Graph TSP Given a connected unweighted graph, a tour is a closed walk that visits every vertex at least once. Objective: find shortest tour. Ideally – a Hamiltonian cycle. 2 3 A tour may use the same edge twice 4 A tour may use the same edge twice length 8x1 + 1x2 = 10 5 Results 1. Size 4n/3 in a graph with a spanning tree and a simple cycle on its odd nodes (WG14, joint with Satoru Iwata and Alantha Newman) “Regular graphs have short tours” 2. Size 9n/7 in cubic bipartite graphs (APPROX14, joint with Jeremy Karp) 1 3. Size (1 + 푂( ))n in d-regular graphs 푑 (IPCO14, joint with Uri Feige and Mohit Singh) 6 Main ideas for short tours 1. Augment spanning tree with carefully chosen edges 2. Delete carefully chosen edges from the whole graph 3. Augment cycle cover with few cycles 4. Augment path cover with few paths 7 General bounds In every connected n-vertex graph, the length of the shortest tour is between n and 2n-2. 8 9 Spanning tree lower bound 10 Double spanning tree edges, drop remaining edges 11 Christofides 1976 • A 3/2-approximation to graph TSP (and more generally, metric TSP). Tour composed of union of: • (minimum) Spanning tree. • Minimum T-join on odd-degree vertices. Gives a connected Eulerian graph (= tour). -
Large Matchings in Bipartite Graphs Have a Rainbow Matching
LARGE MATCHINGS IN BIPARTITE GRAPHS HAVE A RAINBOW MATCHING DANIEL KOTLAR Computer Science Department, Tel-Hai College, Upper Galilee 12210, Israel RAN ZIV Computer Science Department, Tel-Hai College, Upper Galilee 12210, Israel Abstract. Let g(n) be the least number such that every collection of n match- ings, each of size at least g(n), in a bipartite graph, has a full rainbow matching. Aharoni and Berger [1] conjectured that g(n) = n + 1 for every n > 1. This generalizes famous conjectures of Ryser, Brualdi and Stein. Recently, Aharoni, 7 5 Charbit and Howard [2] proved that g(n) ≤ b 4 nc. We prove that g(n) ≤ b 3 nc. 1. Introduction Given sets F1;F2;:::;Fk of edges in a graph a (partial) rainbow matching is a choice of disjoint edges from some of the Fi's. In other words, it is a partial choice function whose range is a matching. If the rainbow matching has disjoint edges from all the Fi's then we call it a full rainbow matching. During the last decade the problem of finding conditions for large rainbow matchings in a graph was extensively explored. See, for example, [1, 8, 10, 11]. Many results and conjectures on the subject were influenced by the well-known conjectures of Ryser [12], asserting that every Latin square of odd order n has a transversal of order n, and Brualdi [6] (see arXiv:1305.1466v2 [math.CO] 21 Oct 2013 also [5] p. 255), asserting that every Latin square of even order n has a partial transversal of size n − 1.