DOCSLIB.ORG

Multiple Circular Colouring As a Model for Scheduling

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multiple Circular Colouring As a Model for Scheduling Open Journal of Discrete Mathematics, 2013, 3, 162-166 http://dx.doi.org/10.4236/ojdm.2013.33029 Published Online July 2013 (http://www.scirp.org/journal/ojdm) Multiple Circular Colouring as a Model for Scheduling Bing Zhou Department of Mathematics, Trent University, Peterborough, Canada Email: [email protected] Received April 30, 2013; revised May 31, 2013; accepted June 15, 2013 Copyright © 2013 Bing Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this article we propose a new model for scheduling periodic tasks. The model is based on a variation of the circular chromatic number, called the multiple circular colouring of the conflict graph. We show that for a large class of graphs, this new model will provide better solutions than the original circular chromatic number. At the same time, it allows us to avoid the difficulty of implementation when the fractional chromatic number is used. Keywords: Graph Coloring; Circular Chromatic Number; Fractional Chromatic Number; Multi-Circular Coloring; Scheduling Problem 1. Introduction edges between them and therefore can be assigned to one period. For this scheduling f, we have f 1 G . We consider the scheduling problems involving tasks Thus in general f 1 G . TT12,,, Tn . If two tasks both use a common resource, they cannot be scheduled at the same time. A valid When the tasks are periodic in nature, circular colour- ing of the conflict graph is a more appropriate method. scheduling is a mapping f from TT12,,, Tn to the subsets of a time period [0,T] such that Circular colouring and the circular chromatic number (also called the star chromatic number) were introduced fTij fT if Ti and Tj use a common re- source. Let the value of a scheduling f be by A. Vince in 1988 [5]. A (k,d)-circular-colouring of a graph G is a mapping cVG:0,1,, k1 such f minfT T :1 i n, which is the minimum i that for each edge xy in G, dcxcykd . length of time a task has been assigned normalized by the The circular chromatic number of G, G , is the in- length of the time period. The goal is to find a scheduling c fimum of the ratio k/d for which G has a (k,d)-circular- f that maximizes f . One example of this type of colouring. scheduling problem is the heavily loaded resource shar- Equivalently, we can consider a (k,d)-circular-colouring ing system in computer science [1-3]. The tasks are proc- of G as a mapping c from V(G) to the open arcs of length esses and some of them may share a common data file. d in a circle of length k such that if xy is an edge in G, Two processes that do share a common data file cannot operate at the same time. A scheduling of the processes cx cy . If we assign the tasks using a (k,d)- that has the maximum value would allow the processes to circular-colouring of the conflict graph G, we would operate most efficiently. have f dk1 c G . The constraints of the scheduling problem can be rep- A (k,d)-set-colouring of a graph G is a mapping c such resented by a graph, called the conflict graph. The con- that for every vertex v in G, c(v) is a d-subset of 0,1, ,k 1 and if xy is an edge in flict graph G has vertex set vv12,,, vn representing G, . The fractional chromatic number of the tasks where two vertices vi and vj are adjacent if and cx cy only if they use at least one common resource. Vertex a graph G, f G , is the infimum of the ratio (k/d) for colouring and chromatic numbers of the conflict graph which G has a (k,d)-set-colouring. If we can consider a have been used as models for scheduling problems (see (k,d)-set-colouring as a mapping c from V(G) to the sets for example [4]). If G can be coloured with kcolours such of arcs in the circle of length k such that the length of the that no two adjacent vertices have the same colour, we union of arcs in f(v) is at least d for every vertex v and if can then divide [0,T] into k equal length periods. All ver- xy is an edge inG, cx cy . An assignment using tices that are coloured with the same colour do not have a (k,d)-set-colouring of the conflict graph G would yield Copyright © 2013 SciRes. OJDM B. ZHOU 163 u1 f dk1 f G . Circular chromatic number and fractional chromatic number and their variations have been extensively stud- v1 ied in the last two decades [6-11]. More results on the u u 5 v v 2 circular chromatic number and fractional chromatic 5 2 number can be found in the book [12] and the survey w v v papers [13,14]. It is easy to see that by their definition, 4 3 we have the relations u u 4 3 fcGGG . Figure 1. Grotzsch graph. Vince proved in [5] that M t Ct52 2. GG1.c f 5 Therefore, we have In comparison, we have GG c . t M Ct5 3. While the difference between G and c G is always less than 1, it is known that the difference be- For this class of graphs, the difference between and f is unbounded. tween G and f G can be arbitrarily large for some graphs. An example is the family of graphs called For the circular chromatic number, we have Kneser graphs. A Kneser graph ([15,16]) K has all tt pq, c M CMC55 12t . q-subsets of 0,1,,p 1 (p > 2q) as its vertices and two vertices are adjacent if the two subsets are disjoint. It This shows that the difference between the circular is proved in [16] that chromatic number and the fractional chromatic number is also arbitrarily large for the Mycielskians. To achieve the p f Kpq,, optimal result for a scheduling problem in general, it q appears that the fractional chromatic number of the con- Kpq,2 p q 2. flict graph provides the best results. However, there are difficulties if we use (k,d)-set col- Another example of a family of graphs where the dif- ouring and the fractional chromatic number in the sched- ference between their chromatic number and fractional uling problem. To achieve f G the optimal (k,d)-set chromatic number is large is the graphs obtained by us- colouring may have a very large value of d. As pointed t ing Mycielski’s construction. out in [12], M C5 is an example of a graph G for For a graph G with vertex set VG uu1,,,2 un which f Gkd for no small d. In fact, if we let and edge set EG, the Mycielskian M G of G is G2 C5 and GMn Gn1 , then the graph with vertex set n1 VG n 32 1 uu12,,, unn vv12 ,,, v w and edge set EuvuuEij,:, ij vwii : 1,2,, n. M C5 in and fnG is a fraction whose denominator, when 2n2 Figure 1 is also called the Grotzsch graph. written in smallest terms, is greater than 2 . This ex- It is well known that ample shows that there is no bound on the denominator of f G that is a polynomial function of the number MG G 1. of vertices of G. If this optimal set colouring is to be used For the fractional chromatic number of Mycielskians, for scheduling, each task would be divided into too many the authors of [17] found the recurrence relation fragments making it impossible in practice. To combine optimality and practicality, in the next section we pro- ffM GG 1.fG pose a new colouring of the conflict graph that will pro- vide a better solution than the circular colouring and eas- For the Grotzsch graph, since C 52, we have f 5 ier to implement than the set colouring. f MC5 52 25 2910. 2. Multiple Circular Colouring of a Graph We use the notation M t G such that Definition 1 An m-(k,d)-circular colouring of a graph G t MG MM( ( MG ( ))). is a mapping c from V(G) to the sets of open arcs in the t times circle of length k such that c(v) is the union of m arcs We have with a total length at least d and if xy is an edge in G, Copyright © 2013 SciRes. OJDM 164 B. ZHOU cx cy . The m-circular chromatic number, m c G is the infimum of the ration kd for which Ghas a m-(k,d)-circular-colouring. w u It is easy to see that for every positive integer m, 3 u2 u 4 u GGGm . 1 fc c w To demonstrate this colouring indeed improves the solution of the scheduling problem in some cases, we 2 show that even for m 2 , c G can be strictly less than c G for large classes of graphs. Recall that for a graph G, M(G) is the Mycielskian of G. There are many graphs G such that c MG MG. Some sufficient conditions for this equality to hold are given in, for example, [18] u5 and [19].
Load more

Recommended publications
  • On J-Colorability of Certain Derived Graph Classes
    On J-Colorability of Certain Derived Graph Classes Federico Fornasiero Department of mathemathic Universidade Federal de Pernambuco Recife, Pernambuco, Brasil [email protected] Sudev Naduvath Centre for Studies in Discrete Mathematics Vidya Academy of Science & Technology Thrissur, Kerala, India. [email protected] Abstract A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every colour class of G with respect to C. A maximal proper colouring of a graph G is a J-colouring of G if and only if every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-colouring of certain Mycielski type graphs. Key Words: Mycielski graphs, graph coloring, rainbow neighbourhoods, J-coloring arXiv:1708.09798v2 [math.GM] 4 Sep 2017 of graphs. Mathematics Subject Classification 2010: 05C15, 05C38, 05C75. 1 Introduction For general notations and concepts in graphs and digraphs we refer to [1, 3, 13]. For further definitions in the theory of graph colouring, see [2, 4]. Unless specified otherwise, all graphs mentioned in this paper are simple, connected and undirected graphs. 1 2 On J-colorability of certain derived graph classes 1.1 Mycielskian of Graphs Let G be a triangle-free graph with the vertex set V (G) = fv1; : : : ; vng. The Myciel- ski graph or the Mycielskian of a graph G, denoted by µ(G), is the graph with ver- tex set V (µ(G)) = fv1; v2; : : : ; vn; u1; u2; : : : ; un; wg such that vivj 2 E(µ(G)) () vivj 2 E(G), viuj 2 E(µ(G)) () vivj 2 E(G) and uiw 2 E(µ(G)) for all i = 1; : : : ; n.
    [Show full text]
  • On Topological Relaxations of Chromatic Conjectures
    European Journal of Combinatorics 31 (2010) 2110–2119 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc On topological relaxations of chromatic conjectures Gábor Simonyi a, Ambrus Zsbán b a Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Hungary b Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary article info a b s t r a c t Article history: There are several famous unsolved conjectures about the chromatic Received 24 February 2010 number that were relaxed and already proven to hold for the Accepted 17 May 2010 fractional chromatic number. We discuss similar relaxations for the Available online 16 July 2010 topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad–Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in Simonyi and Tardos (2006) [41], our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number. ' 2010 Elsevier Ltd. All rights reserved. 1. Introduction There are several hard conjectures about the chromatic number that are still open, while their fractional relaxation is solved, i.e., a similar, but weaker statement is proven for the fractional chromatic number in place of the chromatic number.
    [Show full text]
  • Contemporary Mathematics 352
    CONTEMPORARY MATHEMATICS 352 Graph Colorings Marek Kubale Editor http://dx.doi.org/10.1090/conm/352 Graph Colorings CoNTEMPORARY MATHEMATICS 352 Graph Colorings Marek Kubale Editor American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor Andreas Blass Andy R. Magid Michael Vogeli us This work was originally published in Polish by Wydawnictwa Naukowo-Techniczne under the title "Optymalizacja dyskretna. Modele i metody kolorowania graf6w", © 2002 Wydawnictwa N aukowo-Techniczne. The present translation was created under license for the American Mathematical Society and is published by permission. 2000 Mathematics Subject Classification. Primary 05Cl5. Library of Congress Cataloging-in-Publication Data Optymalizacja dyskretna. English. Graph colorings/ Marek Kubale, editor. p. em.- (Contemporary mathematics, ISSN 0271-4132; 352) Includes bibliographical references and index. ISBN 0-8218-3458-4 (acid-free paper) 1. Graph coloring. I. Kubale, Marek, 1946- II. Title. Ill. Contemporary mathematics (American Mathematical Society); v. 352. QA166 .247.06813 2004 5111.5-dc22 2004046151 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissien@ams.
    [Show full text]
  • On the Generalized Θ-Number and Related Problems for Highly Symmetric Graphs
    On the generalized #-number and related problems for highly symmetric graphs Lennart Sinjorgo ∗ Renata Sotirov y Abstract This paper is an in-depth analysis of the generalized #-number of a graph. The generalized #-number, #k(G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of #k(G), such as that the series (#k(G))k is increasing and bounded above by the order of the graph G. We study #k(G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized #-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs. Keywords k{multicoloring, k-colorable subgraph problem, generalized #-number, Johnson graphs, Hamming graphs, strongly regular graphs. AMS subject classifications. 90C22, 05C15, 90C35 1 Introduction The k{multicoloring of a graph is to assign k distinct colors to each vertex in the graph such that two adjacent vertices are assigned disjoint sets of colors. The k-multicoloring is also known as k-fold coloring, n-tuple coloring or simply multicoloring. We denote by χk(G) the minimum number of colors needed for a valid k{multicoloring of a graph G, and refer to it as the k-th chromatic number of G or the multichromatic number of G.
    [Show full text]
  • Multi-Coloring the Mycielskian of Graphs
    Multi-Coloring the Mycielskian of Graphs Wensong Lin,1 Daphne Der-Fen Liu,2 and Xuding Zhu3,4 1DEPARTMENT OF MATHEMATICS SOUTHEAST UNIVERSITY NANJING 210096, PEOPLE’S REPUBLIC OF CHINA E-mail: [email protected] 2DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY LOS ANGELES, CALIFORNIA 90032 E-mail: [email protected] 3DEPARTMENT OF APPLIED MATHEMATICS NATIONAL SUN YAT-SEN UNIVERSITY KAOHSIUNG 80424, TAIWAN E-mail: [email protected] 4NATIONAL CENTER FOR THEORETICAL SCIENCES TAIWAN Received June 1, 2007; Revised March 24, 2009 Published online 11 May 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jgt.20429 Abstract: A k-fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent Contract grant sponsor: NSFC; Contract grant number: 10671033 (to W.L.); Contract grant sponsor: National Science Foundation; Contract grant number: DMS 0302456 (to D.D.L.); Contract grant sponsor: National Science Council; Contract grant number: NSC95-2115-M-110-013-MY3 (to X.Z.). Journal of Graph Theory ᭧ 2009 Wiley Periodicals, Inc. 311 312 JOURNAL OF GRAPH THEORY vertices are disjoint. The kth chromatic number of a graph G, denoted by k (G), is the minimum total number of colors used in a k-fold coloring of G. Let (G) denote the Mycielskian of G. For any positive integer k,it + ≤ ≤ + holds that k (G) 1 k( (G)) k(G) k (W. Lin, Disc. Math., 308 (2008), 3565–3573). Although both bounds are attainable, it was proved in (Z. Pan, X.
    [Show full text]
  • On B-Colorings and B-Continuity of Graphs
    On b-colorings and b-continuity of graphs By the Faculty of Mathematics and Computer Science of the Technische Universität Bergakademie Freiberg approved Thesis to attain the academic degree doctor rerum naturalium Dr. rer. nat., submitted by Dipl.-Math. Mais Alkhateeb born on January 24, 1979 in Tartous, Syria Referees: Prof. Dr. rer. nat. Ingo Schiermeyer Prof. Dr. rer. nat. Margit Voigt Conferment date: Freiberg, June 28, 2012 To my Home Syria I feel it’s my duty to offer my sincere gratitude to each person who has helped me reach my aim here, either directly or indirectly. I am deeply indebted to my supervisor, Professor Ingo Schiermeyer, and Doctor Anja Kohl for all their help and support. They were there for me whenever I needed them, answered my questions, monitored my progress, helped me and corrected me when I got off the subject. Some shared their knowledge, some shared their experience, some inspired me, but all of them shared the passion I had for this project and made this possible for me. I thank also the Institute of Discrete Mathematics and Algebra, the professors, and all the institute members who gave me the direction and walked with me all along this way. Special thanks go to Professor Margit Voigt for all her interesting suggestions and for refereeing this PhD thesis. I want to remember a few people in my life at this moment, who made this possible for me. I start with my parents who had this vision of providing the best education for their child and created all possible avenues for growth, making numerous personal sacrifices in the process.
    [Show full text]
  • Graph Classes Generated by Mycielskians
    Discussiones Mathematicae Graph Theory 40 (2020) 1163–1173 doi:10.7151/dmgt.2345 GRAPH CLASSES GENERATED BY MYCIELSKIANS Mieczyslaw Borowieckia, Piotr Borowieckib,1 Ewa Drgas-Burchardta and Elzbieta˙ Sidorowicza aInstitute of Mathematics University of Zielona G´ora Prof. Z. Szafrana 4a, 65–516 Zielona G´ora, Poland bFaculty of Electronics, Telecommunications and Informatics Gda´nsk University of Technology Narutowicza 11/12, 80-233 Gda´nsk, Poland e-mail: [email protected] [email protected] [email protected] [email protected] Abstract In this paper we use the classical notion of weak Mycielskian M ′(G) ′ ′ ′ of a graph G and the following sequence: M0(G) = G, M1(G) = M (G), ′ ′ ′ and Mn(G) = M Mn−1(G), to show that if G is a complete graph of order p, then the above sequence is a generator of the class of p-colorable graphs. Similarly, using Mycielskian M(G) we show that analogously de- fined sequence is a generator of the class consisting of graphs for which the chromatic number of the subgraph induced by all vertices that belong to at least one triangle is at most p. We also address the problem of characterizing the latter class in terms of forbidden graphs. Keywords: Mycielski graphs, graph coloring, chromatic number. 2010 Mathematics Subject Classification: 05C15, 05C75, 68R10, 05C60. 1. Introduction The notion of the Mycielski graph (also called the Mycielskian of a graph) was introduced in 1955 by Mycielski [17], which led to the famous construction of 1 Partially supported under Ministry of Science and Higher Education (Poland) subsidy for Gda´nsk University od Technology.
    [Show full text]
  • Degree (Graph Theory) from Wikipedia, the Free Encyclopedia Contents
    Degree (graph theory) From Wikipedia, the free encyclopedia Contents 1 Degree (graph theory) 1 1.1 Handshaking lemma .......................................... 1 1.2 Degree sequence ............................................ 2 1.3 Special values ............................................. 3 1.4 Global properties ........................................... 3 1.5 See also ................................................ 4 1.6 Notes ................................................. 4 1.7 References ............................................... 4 2 Graph operations 5 2.1 Unary operations ........................................... 5 2.1.1 Elementary operations .................................... 5 2.1.2 Advanced operations ..................................... 5 2.2 Binary operations ........................................... 5 2.3 Notes ................................................. 6 3 Regular graph 7 3.1 Existence ............................................... 7 3.2 Algebraic properties .......................................... 7 3.3 Generation .............................................. 8 3.4 See also ................................................ 8 3.5 References .............................................. 8 3.6 External links ............................................. 8 3.7 Text and image sources, contributors, and licenses .......................... 9 3.7.1 Text .............................................. 9 3.7.2 Images ............................................ 9 3.7.3 Content license .......................................
    [Show full text]
  • The Circular Chromatic Number of the Mycielskian Of
    The Circular Chromatic Number of the Mycielskian d of Gk Lingling Huang and Gerard J. Chang∗ DEPARTMENT OF APPLIED MATHEMATICS NATIONAL CHIAO TUNG UNIVERSITY HSINCHU 30050, TAIWAN E-mail: llhuang,[email protected] Received February 2, 1998; revised November 23, 1998 Abstract: In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph µ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G)+1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investi- gated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class d d d of graphs Gk. The main result is that χc(µ(Gk)) = χ(µ(Gk)), which settles a prob- lem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, d k d k to appear]. As χc(Gk)= d and χ(Gk)=d d e, consequently, there exist graphs G such that χc(G) is as close to χ(G) − 1 as you want, but χc(µ(G)) = χ(µ(G)). c 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999 Keywords: chromatic number; circular chromatic number; color; Mycielskian; triangle; clique number 1. INTRODUCTION In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski [9] developed an interesting graph transformation as follows.
    [Show full text]
  • Prime Distance Labeling of Some Families of Graphs
    International Journal of Pure and Applied Mathematics Volume 118 No. 23 2018, 313-321 ISSN: 1314-3395 (on-line version) url: http://acadpubl.eu/hub Special Issue ijpam.eu Prime Distance Labeling of Some Families of Graphs 1, 1 A. Parthiban ∗and N. Gnanamalar David 1 Department of Mathematics, Madras Christian College, Chennai - 600 059, India mathparthi, ngdmcc @gmail.com { } Abstract A graph G is a prime distance graph if there exists a one- to-one labeling of its vertices f : V (G) Z such that for → any two adjacent vertices u and v, the integer f(u) f(v) | − | is a prime. So G is a prime distance graph if and only if there exists a prime distance labeling of G. For a graph G with vertex set V (G) = V and edge set E(G) = E, the Mycielskian of G is the graph µ(G) with vertex set V V ∪ 0 ∪ w , where V = x : x V , and the edge set E xy : { } 0 { 0 ∈ } ∪ { 0 xy E y u : y V . In this paper we establish prime ∈ } ∪ { 0 0 ∈ 0} distance labeling of some Mycielskian graphs and powers of Mycielskian graphs. A graph is said to be triangle-free if no two adjacent vertices are adjacent to a common vertex. In this paper we also consider prime distance labeling for triangle-free graphs and conjecture that every triangle-free graph with χ(G) 4 admits a prime distance labeling. ≤ AMS Subject Classification: 05C15 Key Words and Phrases: Prime Distance Graphs, Prime Distance Labeling, Mycielskian Graphs, Triangle-free Graphs, Coloring, Chromatic Number ∗Research scholar 313 International Journal of Pure and Applied Mathematics Special Issue 1 Introduction All graphs in this paper are simple, finite and undirected.
    [Show full text]
  • Mycielskians and Matchings
    Discussiones Mathematicae 261 Graph Theory 25 (2005) 261{266 MYCIELSKIANS AND MATCHINGS Tomislav Do·slic¶ Department of Informatics and Mathematics Faculty of Agriculture, University of Zagreb Sveto·simunskac. 25, 10000 Zagreb, Croatia e-mail: [email protected] Abstract It is shown in this note that some matching-related properties of graphs, such as their factor-criticality, regularizability and the exis- tence of perfect 2-matchings, are preserved when iterating Mycielski's construction. Keywords: Mycielskian, factor-critical graph, perfect matching, per- fect 2-matching. 2000 Mathematics Subject Classi¯cation: 05C70, 05C75. Members of the sequence of graphs, obtained by starting with K2 and iter- ating Mycielski's construction ([4]), serve as the standard example of graphs whose clique number is much smaller than their chromatic number. Myciel- ski's results were extended in [2], where the fractional chromatic number of iterated Mycielskians was calculated. In the recent article [1] authors consid- ered a number of other properties and graph-theoretic invariants of Myciel- ski's graphs, among others their diameter, Hamiltonicity, biclique partition number, packing number and domination number, as well as some proper- ties of their adjacency matrices. The purpose of this note is to present some results concerning matching-related parameters and structural properties of Mycielski's graphs that have not been considered in previous papers. All graphs considered in this paper will be simple and connected, unless explicitly stated otherwise. For all the terms and concepts not de¯ned here, we refer the reader to the monograph [3]. For a given graph G on the vertex set V (G) = fv1; : : : ; vng, we de- ¯ne its Mycielskian ¹(G) as follows.
    [Show full text]
  • CSCI-761 Combinatorial Computing Assignment 6 Due April 8
    CSCI-761 Combinatorial Computing Assignment 6 due April 8, 2021 Name: Daman Morris 1 1. Pruning Backtracking CNF-SAT is a problem offinding satisfying assignments to a Boolean formulaα in conjunctive normal form, i.e. whereα is given by a set of clauses, each of which is a disjunction of literals, each of which is a variable or a negated variable. Suppose you are solving instances of CNF-SAT by backtracking. Describe informally a bounding function (in the spirit of bounding functions used in the maximum clique search algorithms considered in class) which could be used to prune the search space for satisfying assignments to instances of CNF-SAT. To follow the struggles of many researchers trying to do it better and better already for decades, explore http://www.satcompetition.org, which is a website of worldwide competitions for designing state-of-art SAT-solvers. Since each clause is a disjunction of literals, only one literal needs to be true in order for a clause to be true. In any satisfying assignment, all clauses must be true, so all clausse must have at least one literal that is true. Thus, we can use the number of clauses containing a true literal as a bounding function. In any complete solution this will be equal to the number of clauses, and for any partial assignment X, assigning additional variables will never falsify a clause that is currently true, so this is a true bounding function. 2 2. Ramsey/Folkman numbers (a) Which of the following are true? Give yes/no answer and a brief justification for each.
    [Show full text]