DOCSLIB.ORG

Independence Polynomials of Circulant Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Independence Polynomials of Circulant Graphs INDEPENDENCE POLYNOMIALS OF CIRCULANT GRAPHS by Richard Hoshino SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY AT DALHOUSIE UNIVERSITY HALIFAX, NOVA SCOTIA OCTOBER 15, 2007 c Copyright by Richard Hoshino, 2007 DALHOUSIE UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled \Independence Polynomials of Circulant Graphs" by Richard Hoshino in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dated: October 15, 2007 External Examiner: Dr. Michael Plummer Research Supervisor: Dr. Jason Brown Examining Committee: Dr. Jeannette Janssen Dr. Richard Nowakowski ii DALHOUSIE UNIVERSITY Date: October 15, 2007 Author: Richard Hoshino Title: Independence Polynomials of Circulant Graphs Department: Mathematics and Statistics Degree: Ph.D. Convocation: May Year: 2008 Permission is herewith granted to Dalhousie University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission. The author attests that permission has been obtained for the use of any copyrighted material appearing in the thesis (other than brief excerpts requiring only proper acknowledgement in scholarly writing) and that all such use is clearly acknowledged. iii This is the true joy in life, the being used for a purpose recognized by yourself as a mighty one, the being a force of nature, instead of a selfish, feverish little clod of ailments and grievances complaining that the world will not devote itself to making you happy. I am of the opinion that my life belongs to the whole community, and it is my privilege to do for it whatever I can. Life is no brief candle to me, it is a sort of splendid torch which I've got a hold of for the moment, and I want to make it burn as brightly as possible before handing it on to future generations. { George Bernard Shaw iv Table of Contents List of Tables vii List of Figures viii Abstract ix List of Symbols Used x Acknowledgements xi Chapter 1 Introduction 1 1.1 Basic Terminology . 1 1.2 Circulant Graphs . 2 1.3 The Independence Polynomial . 4 1.4 Overview of the Thesis . 7 Chapter 2 Formulas for Independence Polynomials 9 2.1 The Family S = 1; 2; : : : ; d . 9 f g 2.2 The Family S = d + 1; d + 2; : : : ; n . 15 f b 2 cg 2.3 Circulants of Degree at Most 3 . 31 2.4 Graph Products of Circulants . 40 2.5 Evaluating the Independence Polynomial at x = t . 49 2.6 Independence Unique Graphs . 54 Chapter 3 Analysis of a Recursive Family of Circulant Graphs 63 3.1 Calculating the Independence Number α(Cn;S) . 63 3.2 Application 1: Star Extremal Graphs . 73 3.3 Application 2: Integer Distance Graphs . 107 3.4 Application 3: Fractional Ramsey Numbers . 117 3.5 Application 4: Nordhaus-Gaddum Inequalities . 125 v Chapter 4 Properties and Applications of Circulant Graphs 144 4.1 Line Graphs of Circulants . 144 4.2 List Colourings . 158 4.3 Well-Covered Circulants . 171 4.4 Independence Complexes of Circulant Graphs . 200 Chapter 5 Roots of Independence Polynomials 220 5.1 The Real Roots of I(Cn;S; x) . 222 5.2 The Root of Minimum Modulus of I(Cn;S; x) . 226 5.3 The Root of Maximum Modulus of I(Cn;S; x) . 236 5.4 The Rational Roots of I(Cn;S; x) . 240 5.5 The Roots of Independence Polynomials and their Closures . 247 Chapter 6 Conclusion 254 Bibliography 258 vi List of Tables Table 3.1 The nine known Ramsey numbers of the form r(a; b). 119 Table 4.1 Eulerian subgraphs for the circulant C9; 1;2 . 161 f g Table 4.2 Connected well-covered circulants on at most 12 vertices. 172 Table 4.3 Connected well-covered circulants with G 12 and α(G) 3. 219 j j ≤ ≥ Table 4.4 Shelling of the pure complex ∆(C12; 1;3;6 ). 219 f g Table 5.1 Examples of graphs for which I(Bn; x) is not stable. 225 Table 5.2 The closures of the roots of graph polynomials. 253 vii List of Figures Figure 1.1 The circulant graphs C9; 1;2 and C9; 3;4 . 3 f g f g Figure 1.2 Two independent sets of size 3 in C6. 4 Figure 2.1 The ladder graph Un. 32 Figure 2.2 The graphs G9 = C18; 1;9 and H9 = C18; 2;9 . 32 f g f g Figure 2.3 The graphs G = C18; 2;9 and H = C18; 8;9 . 35 f g f g Figure 2.4 The graph products C 2C and C C . 41 3 3 3 × 3 Figure 2.5 Two trees with the same independence polynomial. 55 Figure 2.6 The graphs G = C8; 3;4 and the corresponding graph H3. 57 f g 5 Figure 3.1 A 3-colouring of C5 and a fractional 2 -colouring of C5. 74 Figure 4.1 Possible subgraphs of G induced by these 7 edges. 149 Figure 4.2 Possible subgraphs of G induced by e0; e1; e2; e3; en 1 . 150 f − g Figure 4.3 Two possible edge labellings of K4. 153 Figure 4.4 Two possible K subgraphs induced by e ; e ; e ; e ; e ; e : 155 4 f 0 a 2a 3a 4a 5ag Figure 4.5 Two possible subgraphs induced by this set of eight edges. 157 Figure 4.6 A 2-list colouring of K2;4 that is not proper. 159 Figure 4.7 A 3-list colouring of K7;7 that is not proper. 160 Figure 4.8 Graphs which generate the family of cubic graphs. 177 Figure 4.9 Four well-covered cubic graphs. 178 Figure 4.10 Graph X and its vertices labelled. 179 Figure 4.11 Graph Y and its vertices labelled. 180 Figure 4.12 Graph Z and its vertices labelled. 181 Figure 4.13 Separating nk = aknk 1 1 vertices into ak classes. 189 − − Figure 4.14 The independence complex ∆(C6). 202 Figure 4.15 The independence complex ∆(3K2). 202 viii Abstract The circulant graph C is the graph on n vertices (with labels 0; 1; 2; : : : ; n 1), n;S − spread around a circle, where two vertices u and v are adjacent iff their (minimum) distance u v appears in set S. In this thesis, we provide a comprehensive analysis of j − j the independence polynomial I(G; x), when G = Cn;S is a circulant. The independence polynomial is the generating function of the number of independent sets of G with k vertices, for k 0. ≥ While it is NP -hard to determine the independence polynomial I(G; x) of an arbitrary graph G, we are able to determine explicit formulas for I(G; x) for several families of circulants, using techniques from combinatorial enumeration. We then describe a recursive construction for an infinite family of circulants, and determine the independence number of each graph in this family. We use these results to provide four applications, encompassing diverse areas of discrete mathematics. First, we determine a new (infinite) family of star extremal graphs. Secondly, we obtain a formula for the chromatic number of infinitely many integer distance graphs. Thirdly, we prove an explicit formula for the generalized fractional Ramsey function. Finally, we determine the optimal Nordhaus-Gaddum inequalities for the fractional chromatic and circular chromatic numbers. These new theorems significantly extend what is currently known. Building on these results, we develop additional properties and applications of circulant graphs. We determine a full characterization of all graphs G for which its line graph L(G) is a circulant, and apply our previous theorems to calculate the list colouring number of a specific family of circulants. We then characterize well-covered circulant graphs, and examine the shellability of their independence complexes. We conclude the thesis with a detailed analysis of the roots of I(Cn;S; x). Among many other results, we solve several open problems by calculating the density of the roots of these independence polynomials, leading to new theorems on the roots of matching polynomials and rook polynomials. ix Abstract Cn;S The circulant graph on n vertices with generating set S I(G; x) The independence polynomial of graph G Pn The path on n vertices Cn The cycle on n vertices Kn The complete graph on n vertices G The order of a graph G, i.e., the number of vertices in G j j u v The circular distance of two vertices u and v in C j − jn n;S deg(v) The degree of a vertex v α(G) The independence number of G !(G) The clique number of a G χ(G) The chromatic number of G G The complement of G L(G) The line graph of G [xk]P (x) The coefficient of the xk term of P (x) An The circulant Cn; 1;2;:::;d , where d 1 is fixed f g ≥ Bn The circulant Cn; d+1;d+2;:::; n , where d 0 is fixed f b 2 cg ≥ G[H] The lexicographic graph product of G with H χf (G) The fractional chromatic number of G χc(G) The circular chromatic number of G !f (G) The fractional clique number of G χl(G) The list colouring number of G G(Z; S) The integer distance graph generated by set S ρ(G) The vertex arboricity number of G r(a1; a2; : : : ; ak) The Ramsey function of the k-tuple (a1; a2; : : : ; ak) r!f (a1; a2; : : : ; ak) The fractional Ramsey function of the k-tuple (a1; a2; : : : ; ak) ∆(G) The independence complex of G x Acknowledgements First and foremost, I'd like to thank my supervisor Jason Brown, for his incredible support and patience during the completion of this thesis.
Load more

Recommended publications
  • Connectivities and Diameters of Circulant Graphs
    CONNECTIVITIES AND DIAMETERS OF CIRCULANT GRAPHS Paul Theo Meijer B.Sc. (Honors), Simon Fraser University, 1987 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (MATHEMATICS) in the Department of Mathematics and Statistics @ Paul Theo Meijer 1991 SIMON FRASER UNIVERSITY December 1991 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. Approval Name: Paul Theo Meijer Degree: Master of Science (Mathematics) Title of Thesis: Connectivities and Diameters of Circulant Graphs Examining Committee: Chairman: Dr. Alistair Lachlan Dr. grian Alspach, Professor ' Senior Supervisor Dr. Luis Goddyn, Assistant Professor - ph Aters, Associate Professor . Dr. Tom Brown, Professor External Examiner Date Approved: December 4 P 1991 PART IAL COPYH IGIiT L ICLNSI: . , I hereby grant to Sirnori Fraser- llr~ivorsitytho righl to lend my thesis, project or extended essay (tho title of which is shown below) to users of the Simon Frasor University Libr~ry,and to make part ial or single copies only for such users or in response to a request from the library of any other university, or other educational insfitution, on 'its own behalf or for one of its users. I further agree that percnission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission. Title of Thesis/Project/Extended Essay (date) Abstract Let S = {al, az, .
    [Show full text]
  • Graph Construction Using the Dk-Series Framework
    UNIVERSITY OF CALIFORNIA, IRVINE Graph Construction using the dK-Series Framework DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Networked Systems by Balint Tillman Dissertation Committee: Professor and Chancellor's Fellow Athina Markopoulou, Chair Chancellor's Professor David Eppstein Professor Carter T. Butts 2019 Portion of Chapter 2 and Chapter 3 c 2019 IEEE All other materials c 2019 Balint Tillman DEDICATION To my friends and family who were there to support me through this work and helped me to overcome obstacles along the way. ii TABLE OF CONTENTS Page LIST OF FIGURES v LIST OF TABLES ix LIST OF ALGORITHMS x ACKNOWLEDGMENTS xi CURRICULUM VITAE xii ABSTRACT OF THE DISSERTATION xiii 1 Introduction 1 1.1 Motivation . .1 1.2 Problem Statement . .3 1.3 Our Work in Perspective . .5 1.3.1 Prior Work on Undirected Graph Construction . .5 1.3.2 Prior Work on Directed Graph Construction . .9 1.3.3 Dissertation Contributions: The 2K+ Framework . 10 2 Undirected Graph Construction 16 2.1 Introduction . 16 2.2 2K Construction: JDM . 17 2.2.1 Realizability . 17 2.2.2 Algorithm for 2K Construction . 18 2.2.3 Connections to Related Work . 24 2.2.4 Space of Realizations . 25 2.3 2K with additional constraints . 27 2.3.1 2K+S: Target JDM and Clustering . 27 2.3.2 2K+#4: NP-Hardness for JDM with fixed number of triangles . 33 2.3.3 2K+A: Targeting JDM and Node Attributes . 38 2.3.4 2K+CC: Number of Connected Components .
    [Show full text]
  • Distance Labelings of Möbius Ladders
    Distance Labelings of M¨obiusLadders A Major Qualifying Project Report: Submitted to the Faculty of WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Bachelor of Science by Anthony Rojas Kyle Diaz Date: March 12th; 2013 Approved: Professor Peter R. Christopher Abstract A distance-two labeling of a graph G is a function f : V (G) ! f0; 1; 2; : : : ; kg such that jf(u) − f(v)j ≥ 1 if d(u; v) = 2 and jf(u) − f(v)j ≥ 2 if d(u; v) = 1 for all u; v 2 V (G). A labeling is optimal if k is the least possible integer such that G admits a k-labeling. The λ2;1 number is the largest integer assigned to some vertex in an optimally labeled network. In this paper, we examine the λ2;1 number for M¨obiusladders, a class of graphs originally defined by Richard Guy and Frank Harary [9]. We completely determine the λ2;1 number for M¨obius ladders of even order, and for a specific class of M¨obiusladders with odd order. A general upper bound for λ2;1(G) is known [6], and for the remaining cases of M¨obiusladders we improve this bound from 18 to 7. We also provide some results for radio labelings and extensions to other labelings of these graphs. Executive Summary A graph is a pair G = (V; E), such that V (G) is the vertex set, and E(G) is the set of edges. For simple graphs (i.e., undirected, loopless, and finite), the concept of a radio labeling was first introduced in 1980 by Hale [8].
    [Show full text]
  • THE CRITICAL GROUP of a LINE GRAPH: the BIPARTITE CASE Contents 1. Introduction 2 2. Preliminaries 2 2.1. the Graph Laplacian 2
    THE CRITICAL GROUP OF A LINE GRAPH: THE BIPARTITE CASE JOHN MACHACEK Abstract. The critical group K(G) of a graph G is a finite abelian group whose order is the number of spanning forests of the graph. Here we investigate the relationship between the critical group of a regular bipartite graph G and its line graph line G. The relationship between the two is known completely for regular nonbipartite graphs. We compute the critical group of a graph closely related to the complete bipartite graph and the critical group of its line graph. We also discuss general theory for the critical group of regular bipartite graphs. We close with various examples demonstrating what we have observed through experimentation. The problem of classifying the the relationship between K(G) and K(line G) for regular bipartite graphs remains open. Contents 1. Introduction 2 2. Preliminaries 2 2.1. The graph Laplacian 2 2.2. Theory of lattices 3 2.3. The line graph and edge subdivision graph 3 2.4. Circulant graphs 5 2.5. Smith normal form and matrices 6 3. Matrix reductions 6 4. Some specific regular bipartite graphs 9 4.1. The almost complete bipartite graph 9 4.2. Bipartite circulant graphs 11 5. A few general results 11 5.1. The quotient group 11 5.2. Perfect matchings 12 6. Looking forward 13 6.1. Odd primes 13 6.2. The prime 2 14 6.3. Example exact sequences 15 References 17 Date: December 14, 2011. A special thanks to Dr. Victor Reiner for his guidance and suggestions in completing this work.
    [Show full text]
  • ABC Index on Subdivision Graphs and Line Graphs
    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728 p-ISSN: 2319–765X PP 01-06 www.iosrjournals.org ABC index on subdivision graphs and line graphs A. R. Bindusree1, V. Lokesha2 and P. S. Ranjini3 1Department of Management Studies, Sree Narayana Gurukulam College of Engineering, Kolenchery, Ernakulam-682 311, Kerala, India 2PG Department of Mathematics, VSK University, Bellary, Karnataka, India-583104 3Department of Mathematics, Don Bosco Institute of Technology, Bangalore-61, India, Recently introduced Atom-bond connectivity index (ABC Index) is defined as d d 2 ABC(G) = i j , where and are the degrees of vertices and respectively. In this di d j vi v j di .d j paper we present the ABC index of subdivision graphs of some connected graphs.We also provide the ABC index of the line graphs of some subdivision graphs. AMS Subject Classification 2000: 5C 20 Keywords: Atom-bond connectivity(ABC) index, Subdivision graph, Line graph, Helm graph, Ladder graph, Lollipop graph. 1 Introduction and Terminologies Topological indices have a prominent place in Chemistry, Pharmacology etc.[9] The recently introduced Atom-bond connectivity (ABC) index has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. Furtula et al. (2009) [4] obtained extremal ABC values for chemical trees, and also, it has been shown that the star K1,n1 , has the maximal ABC value of trees. In 2010, Kinkar Ch Das present the lower and upper bounds on ABC index of graphs and trees, and characterize graphs for which these bounds are best possible[1].
    [Show full text]
  • Arxiv:1802.04921V2 [Math.CO]
    STABILITY OF CIRCULANT GRAPHS YAN-LI QIN, BINZHOU XIA, AND SANMING ZHOU Abstract. The canonical double cover D(Γ) of a graph Γ is the direct product of Γ and K2. If Aut(D(Γ)) = Aut(Γ) × Z2 then Γ is called stable; otherwise Γ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arc- transitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Maruˇsiˇc, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices. Key words: circulant graph; stable graph; compatible adjacency matrix 1. Introduction We study the stability of circulant graphs. Among others we answer a question of Wilson [14] and give infinitely many counterexamples to a conjecture of Maruˇsiˇc, Scapellato and Zagaglia Salvi [9]. All graphs considered in the paper are finite, simple and undirected. As usual, for a graph Γ we use V (Γ), E(Γ) and Aut(Γ) to denote its vertex set, edge set and automorphism group, respectively. For an integer n > 1, we use nΓ to denote the graph consisting of n vertex-disjoint copies of Γ. The complete graph on n > 1 vertices is denoted by Kn, and the cycle of length n > 3 is denoted by Cn. In this paper, we assume that each symbol representing a group or a graph actually represents the isomorphism class of the same.
    [Show full text]
  • Cores of Vertex-Transitive Graphs
    Cores of Vertex-Transitive Graphs Ricky Rotheram Submitted in total fulfilment of the requirements of the degree of Master of Philosophy October 2013 Department of Mathematics and Statistics The University of Melbourne Parkville, VIC 3010, Australia Produced on archival quality paper Abstract The core of a graph Γ is the smallest graph Γ∗ for which there exist graph homomor- phisms Γ ! Γ∗ and Γ∗ ! Γ. Thus cores are fundamental to our understanding of general graph homomorphisms. It is known that for a vertex-transitive graph Γ, Γ∗ is vertex-transitive, and that jV (Γ∗)j divides jV (Γ)j. The purpose of this thesis is to determine the cores of various families of vertex-transitive and symmetric graphs. We focus primarily on finding the cores of imprimitive symmetric graphs of order pq, where p < q are primes. We choose to investigate these graphs because their cores must be symmetric graphs with jV (Γ∗)j = p or q. These graphs have been completely classified, and are split into three broad families, namely the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. We use this classification to determine the cores of all imprimitive symmetric graphs of order pq, using differ- ent approaches for the circulants, the incidence graphs and the Maruˇsiˇc-Scapellato graphs. Circulant graphs are examples of Cayley graphs of abelian groups. Thus, we generalise the approach used to determine the cores of the symmetric circulants of order pq, and apply it to other Cayley graphs of abelian groups. Doing this, we show that if Γ is a Cayley graph of an abelian group, then Aut(Γ∗) contains a transitive subgroup generated by semiregular automorphisms, and either Γ∗ is an odd cycle or girth(Γ∗) ≤ 4.
    [Show full text]
  • Integral Circulant Graphsଁ Wasin So
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 306 (2005) 153–158 www.elsevier.com/locate/disc Note Integral circulant graphsଁ Wasin So Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA Received 10 September 2003; received in revised form 6 September 2005; accepted 17 November 2005 Available online 4 January 2006 Abstract − In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2(n) 1 non-isomorphic integral circulant graphs on n vertices, where (n) is the number of divisors of n. © 2005 Elsevier B.V. All rights reserved. Keywords: Integral graph; Circulant graph; Graph spectrum 1. Graph spectra Let G be a simple graph which is a graph without loop, multi-edge, and orientation. Denote A(G) the adjacency matrix of G with respect to a given labeling of its vertices. Note that A(G) is a symmetric (0, 1)-matrix with zero diagonal entries. Adjacency matrices with respect to different labeling are permutationally similar. The spectrum of G, sp(G), is defined as the spectrum of A(G), which is the set of all eigenvalues of A(G) including multiplicity. Since the spectrum of a matrix is invariant under permutation similarity, sp(G) is independent of the labeling used in A(G). Because A(G) is a real symmetric matrix, sp(G) contains real eigenvalues only. Moreover, A(G) is an integral matrix and so its characteristic polynomial is monic and integral, hence its rational roots are integers.
    [Show full text]
  • Graceful Labeling of Some New Graphs ∗
    Bulletin of Pure and Applied Sciences Bull. Pure Appl. Sci. Sect. E Math. Stat. Section - E - Mathematics & Statistics 38E(Special Issue)(2S), 60–64 (2019) e-ISSN:2320-3226, Print ISSN:0970-6577 Website : https : //www.bpasjournals.com/ DOI: 10.5958/2320-3226.2019.00080.8 c Dr. A.K. Sharma, BPAS PUBLICATIONS, 387-RPS- DDA Flat, Mansarover Park, Shahdara, Delhi-110032, India. 2019 Graceful labeling of some new graphs ∗ J. Jeba Jesintha1, K. Subashini2 and J.R. Rashmi Beula3 1,3. P.G. Department of Mathematics, Women’s Christian College, Affiliated to University of Madras, Chennai-600008, Tamil Nadu, India. 2. Research Scholar (Part-Time), P.G. Department of Mathematics, Women’s Christian College, Affiliated to University of Madras, Chennai-600008, Tamil Nadu, India. 1. E-mail: jjesintha [email protected] , 2. E-mail: [email protected] Abstract A graceful labeling of a graph G with q edges is an injection f : V (G) → {0, 1, 2,...,q} with the property that the resulting edge labels are distinct where the edge incident with the vertices u and v is assigned the label |f (u) − f (v) |. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove that the series of isomorphic copies of Star graph connected between two Ladders are graceful. Key words Graceful labeling, Path, Ladder Graph, Star Graph. 2010 Mathematics Subject Classification 05C60, 05C78. 1 Introduction In 1967, Rosa [2] introduced the graceful labeling method as a tool to attack the Ringel-Kotzig-Rosa Conjecture or the Graceful Tree Conjecture that “All Trees are Graceful” and he also proved that caterpillars (a caterpillar is a tree with the property that the removal of its endpoints leaves a path) are graceful.
    [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]
  • Ricci Curvature, Circulants, and a Matching Condition 1
    RICCI CURVATURE, CIRCULANTS, AND A MATCHING CONDITION JONATHAN D. H. SMITH Abstract. The Ricci curvature of graphs, as recently introduced by Lin, Lu, and Yau following a general concept due to Ollivier, provides a new and promising isomorphism invariant. This pa- per presents a simplified exposition of the concept, including the so-called logistic diagram as a computational or visualization aid. Two new infinite classes of graphs with positive Ricci curvature are identified. A local graph-theoretical condition, known as the matching condition, provides a general formula for Ricci curva- tures. The paper initiates a longer-term program of classifying the Ricci curvatures of circulant graphs. Aspects of this program may prove useful in tackling the problem of showing when twisted tori are not isomorphic to circulants. 1. Introduction Various analogues of Ricci curvature have been extended from the context of Riemannian manifolds in differential geometry to more gen- eral metric spaces [1, 11, 12], and in particular to connected locally finite graphs [2, 3, 5, 6, 8, 9]. In graph theory, the concept appears to be a significant isomorphism invariant (in addition to other uses such as those discussed in [8, x4]). A typical problem where it may be of interest concerns the class of valency 4 graphs known as (rectangular) twisted tori, which have gained attention through their applications to computer architecture [4]. The apparently difficult open problem is to prove that, with finitely many exceptions, these graphs are not circu- lant (since this would show that they provide a genuinely new range of architectures). Now it is known that the twisted tori are quotients of cartesian products of cycles.
    [Show full text]
  • Total Coloring Conjecture for Certain Classes of Graphs
    algorithms Article Total Coloring Conjecture for Certain Classes of Graphs R. Vignesh∗ , J. Geetha and K. Somasundaram Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore 641112, India; [email protected] (J.G.); [email protected] (K.S.) * Correspondence: [email protected] Received: 30 August 2018; Accepted: 17 October 2018; Published: 19 October 2018 Abstract: A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by c00(G), is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, D(G) + 1 ≤ c00(G) ≤ D(G) + 2, where D(G) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph. Keywords: total coloring; lexicographic product; deleted lexicographic product; line graph; double graph MSC: 05C15 1. Introduction All the graphs in this paper are finite, simple and connected. The edge chromatic number of a graph G, denoted by c0(G) , is the smallest number of colors needed to color the edges of G so that no two adjacent edges share the same color. For any graph G, it clear that from the Vizing’s theorem that the edge chromatic number c0(G) ≤ D(G) + 1, where D(G) is the maximum degree of G. If c0(G) = D(G) then G is called class-I graph and if c0(G) = D(G) + 1 then G is called class-II graph.
    [Show full text]