Zachary Monaco Georgia College Olympic Coloring: Go For The Gold
|
|
- Augustus Basil Shaw
- 7 years ago
- Views:
Transcription
1 Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various scheduling questions and other optimization problems For this paper, we consider the idea of edge-coloring a graph, which is a coloring of the edges such that adjacent edges are distinctly colored The minimal number of colors required to edge-color a graph is called the chromatic index This paper discusses the chromatic indices of graphs obtained from certain symmetric matrices 1 Introduction The goal of this paper is to determine the chromatic index of a graph given by a certain family of matrices The family that we are going to be working with are called Golden matrices The name comes from their connection with the golden fields described by Peter Steinbach which are further explained [FH11] In this paper, we discuss the adjacency matrices of these Golden matrices and go through different cases to find a connection between the dimension of its associated matrix and the chromatic index of the graph 2 Preliminaries In this section, we will identify terms and notation that will be used throughout this paper Any graph G consists of the non-empty set of elements called vertices and a set consisting of ordered pairs (u, v), where u and v are vertices, that are called edges Let the degree of a vertex v, denoted deg(v), be the number of edges incident with this vertex where a loop contributes two to the degree of v Adjacent edges are edges that share a common vertex and adjacent vertices are vertices that are connected by at least one edge Let a loop be an edge that connects a vertex to itself A graph has multiple edges if a vertex v has more than one edge connection to the another vertex We define a simple graph as a graph that contains no loops or multiple edges Throughout the paper we will be referring to the chromatic index of a graph, denoted χ (G), which is the number of distinct colors required to edge-color a graph where the coloring of the edges requires that no two adjacent edges share the same color If we say our graph G is 1
2 2 disconnected that means it is possible to express the graph as a union of two subgraphs and G is connected if it cannot be so expressed See [Wil10] for more information about graphs Our graphs involve unions of certain families of basic graphs As such, we introduce some notation for these graphs Let C n represent a cycle graph on n vertices where the degree of each vertex is 2 Let P n represent a path graph on n vertices where P n is obtained by removing an edge from C n The graphs in our paper will be constructed using a family of matrices called Golden Matrices Golden Matrices are constructed using the following definition: let n 3 N, k = 1,, n 1, and let G[n, k] be the square matrix with dimension n 1 given by G[n, k] = (a u,v ) where a u,v = 1, if v = k u + 2i 1 for some positive integer i min{u, k, n u, n k}; a u,v = 0, otherwise These Golden Matrices serve as the adjacency matrices for the graphs we will consider 3 G[n, 1] and G[n, n 1] Proposition 31 For all n 3, χ (G[n, 1]) = 1 Proof Let G[n, 1] be our given matrix This implies that k = 1 which further implies that i = 1 because i =min {u, k, n u, n k} and k = 1 Using the defined values for k and i, we can solve for v Observe that v = 1 u + 2(1) 1 = 1 u = 1 u + 1 = 1 + u + 1 = u Then using the definition of the adjacency matrix for G[n, k], G[n, 1] = (a u,v ) where { a u,v = 1, if v = u; a u,v = 0, otherwise
3 Thus we have a value of 1 wherever a u,v when u = v Hence, the corresponding adjacency matrix is Note that in each row and column there is exactly one entry at a u,v where u = v Thus we know each vertex is connected only to itself with a loop creating n disjoint loops Since all n loops are disjoint, we can use one color to edge-color the graph Hence χ (G[n, 1]) = 1 for all n 3 Now we will consider the k = n 1 case Proposition 32 For all n 3, χ (G[n, n 1]) = 1 Proof Let G[n, n 1] be our given matrix This implies that k = n 1 and it follows that i = 1 since i =min {u, k, n u, n k} and n k = n (n 1) = 1 Then using u n 1 and the definition of the adjacency matrices, we have Thus, v = (n 1) u + 2(1) 1 = n 1 u = n u G[n, n 1] = (a u,v ) where { a u,v = 1, if v = n u; a u,v = 0, otherwise, has the following adjacency matrix: From the adjacency matrix we will denote the following edge pairings
4 4 as u v, where u is the row and v are the position of the one entries The corresponding edge pairing from our matrix are 1 n 1, 2 n 2, 3 n 3, n n n n We know that we stop at because our matrix is square with dimension (n 1) and each entry will have one pair 2 We will show that the adjacency matrix of G[n, n 1] can give us two different cases when n is even or n is odd, but in both cases we will have χ (G[n, n 1]) = 1 Let n be even Then by definition of even, n = 2l for some l Z If we substitute this value for n into our pairing system observe that n 2l 2l n = = l l = 2l = Thus we have l l which creates a loop in our graph Thus our pairings and our adjacency matrix show that we will create multiple disjoint line segments and one disjoint loop Since each line segment and the loop are all disjoint, we say that χ (G[n, n 1]) = 1 if n is even Let n be odd Then by definition of even n = 2l + 1 for some l Z If we substitute this value for n into our pairing system this yields n 2l + 1 2l + 1 n = = l l + 1 = 2l + 1 = This yields the second graph we obtain which is multiple disjoint line segments Note all line segments are disjoint and we deduce that χ (G[n, n 1]) = 1, when n is odd Consequently, χ (G([n, n 1])) = 1 for all n 3 4 G[n, 2] and G[n, n 2] Now that we ahve dealt with the k = 1 and k = n 1 case, let us talk about the cases k = 2 nd k = n 2
5 5 Proposition 41 For all n 3, χ (G[n, 2]) = 2 Proof Note by the nature of the value of i in our equation to find v, we will end up having 4 different cases First we will break up the cases into when i = 1 and when i = 2 Then our subcases will be when u > 2 or when u 2 due to the absolute value of k u being either negative or non-negative So let i = 1 Then when we solve for v we get: if u > 2, and if u 2, v = 2 u = 2 u + 1 = 2 + u + 1 = u 1, v = 2 u + 1 = 3 u Next we look at when i = 2 Observe the following values for v: if u > 2, and if u 2, v = 2 u = 2 u + 3 = 2 + u + 3 = u + 1, v = 2 u + 3 = 5 u Now we will use this information to construct our adjacency matrix for G[n, 2] Observe that now we must use the cases when i = 1 and i = 2 to complete our matrix Note that we must complete all the cases for when i = 1 to obtain the following edge pairings for u v: 1 2, 2 1, 3 3, 4 3, (n 1) (n 2) This creates a diagonal that runs from a 2,1 to a n 1,n 2 Now we must do the edge pairings for when i = 2 Using the apporpriate equations, we obtain (n 2) (n 1)
6 6 This creates a diagonal that runs from a 1,2 to a n 2,n 1 Putting this information together we obtain an adjacency matrix that looks like These two diagonals in our adjacency matrix create a path graph and we know that χ (P n ) = 2 Hence we have χ (G[n, 2]) = 2 Proposition 42 For all n 3, χ (G[n, n 2]) = 2 Proof To begin we find the possible values for i that will be used in calculating v We know that i min{u, k, n u, n k} Thus we plug in our given values for n and k and we see that i min{u, n 2, n u, 2} Hence, i 2 So since k = n 2, we say that when u = 1 or u = n 1 v = n = n 2, when u = 1, and v = (n 2) (n 1) + 1 = = 2, when u = n 1 Then when u = 2, 3, 4,, n 2, we must consider the cases when i = 1 and i = 2 First let i = 1 This yields v = n 2 u = n 2 u + 1 = n u 1 Next we check the values for v when i = 2 Given that i = 2, we solve v to get v = n 2 u = n 2 u + 3 = n u + 1 By calculating each v with each corresponding value of u, we deduce the following edge pairings of u v 1 n 2, 2 n 3, n 1, 3 n 4, n 2, n 2 1, 3, n 1 2
7 7 From these edge pairing we can produce the adjacency matrix below We know that the given graph for the matrix depends on when n is even or odd When n is even, we have a path graph which we have already shown has chromatic index of 2 Lastly when n is odd, we have two identical path graphs with loops on the endpoints Thus we can color the path graph with 2 colors and the loop can be colored with one of the remaining two colors Hence we have χ (G[n, n 2]) = 2 for all n N What we began to create was a pyramid of values that represent the chromatic indices based on the row and position in the row The row in the pyramid represents the n and the position represents k in G[n, k] We proved the boundary cases for k = 1, k = n 1, k = 2, and k = n 2 for all n N The pyramid is presented below along with our conjecture based on this pyramid of chromatic indices Conjecture 43 The formula for finding χ (G[n, k]) depends on when n is even or n is odd If n is odd, then {χ (G[n, 1]), χ (G[n, 2]), χ (G[n, 3])χ (G[n, n 2]), χ (G[n, n 1])} = {1, 2, 3,, ( n 2 1), ( n) (n) (n,, 2 1),, 3, 2, 1},
8 8 where the position of the index represents k in G[n, k] If n is even, then {χ (G[n, 1]), χ (G[n, 2]), χ (G[n, 3])χ (G[n, n 2]), χ (G[n, n 1])} = {1, 2, 3,, ( n ), ( n + 1) (n + 1, 1 ),, 3, 2, 1}, 2 where the position of the index represents k in G[n, k] 5 Conclusion In this paper we have found the chromatic indices for the first two boundary cases of G[n, 1], G[n, n 1], G[n, 2], and G[n, n 2] This method of proving the chromatic indices would not be sufficient for all n N since we know the set of natrual numbers is unbounded With this we would need to find a proof for the general case made in the conjecture, but this is a topic for further research We began to make connections between an algebraic family of matrices and graph theory and another topic that can be researched into more depth would be if the algebraic properties such as the product-sum formula holds while using the chromatic indices that may be found from the conjecture found in this paper References [FH11] Anne Fontaine and Susan Hurley Golden matrix families The College Mathematics Journal, 42(2): , 2011 [Wil10] Robin James Wilson Introduction to Graph Theory, 5/e Pearson Education, 2010
On Integer Additive Set-Indexers of Graphs
On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationA permutation can also be represented by describing its cycles. What do you suppose is meant by this?
Shuffling, Cycles, and Matrices Warm up problem. Eight people stand in a line. From left to right their positions are numbered,,,... 8. The eight people then change places according to THE RULE which directs
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 10: Graphs
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph
More informationSECTIONS 1.5-1.6 NOTES ON GRAPH THEORY NOTATION AND ITS USE IN THE STUDY OF SPARSE SYMMETRIC MATRICES
SECIONS.5-.6 NOES ON GRPH HEORY NOION ND IS USE IN HE SUDY OF SPRSE SYMMERIC MRICES graph G ( X, E) consists of a finite set of nodes or vertices X and edges E. EXMPLE : road map of part of British Columbia
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationSocial Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
More informationClass One: Degree Sequences
Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of
More informationGraphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationExponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics
More informationDiscrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
More informationDETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationM-Degrees of Quadrangle-Free Planar Graphs
M-Degrees of Quadrangle-Free Planar Graphs Oleg V. Borodin, 1 Alexandr V. Kostochka, 1,2 Naeem N. Sheikh, 2 and Gexin Yu 3 1 SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK 630090, RUSSIA E-mail: brdnoleg@math.nsc.ru
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More information(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7
(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition
More informationThe chromatic spectrum of mixed hypergraphs
The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex
More informationClique coloring B 1 -EPG graphs
Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La
More informationDeterminants in the Kronecker product of matrices: The incidence matrix of a complete graph
FPSAC 2009 DMTCS proc (subm), by the authors, 1 10 Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph Christopher R H Hanusa 1 and Thomas Zaslavsky 2 1 Department
More informationSHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE
SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless
More informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
More informationExamination paper for MA0301 Elementær diskret matematikk
Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationOdd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationA 2-factor in which each cycle has long length in claw-free graphs
A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationMinimum rank of graphs that allow loops. Rana Catherine Mikkelson. A dissertation submitted to the graduate faculty
Minimum rank of graphs that allow loops by Rana Catherine Mikkelson A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major:
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More informationCycles and clique-minors in expanders
Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor
More informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationarxiv:math/0606467v2 [math.co] 5 Jul 2006
A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group arxiv:math/0606467v [math.co] 5 Jul 006 Richard P. Stanley Department of Mathematics, Massachusetts
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationRemoving Even Crossings
EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationJust the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
More informationAn inequality for the group chromatic number of a graph
Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More information8. Matchings and Factors
8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,
More informationTechnology, Kolkata, INDIA, pal.sanjaykumar@gmail.com. sssarma2001@yahoo.com
Sanjay Kumar Pal 1 and Samar Sen Sarma 2 1 Department of Computer Science & Applications, NSHM College of Management & Technology, Kolkata, INDIA, pal.sanjaykumar@gmail.com 2 Department of Computer Science
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationSEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH
CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationUPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE
UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMidterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
More informationOn an anti-ramsey type result
On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element
More informationOn the crossing number of K m,n
On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known
More informationCS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin
CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework
More informationMinimum degree condition forcing complete graph immersion
Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationTypes of Degrees in Bipolar Fuzzy Graphs
pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 4857-4866 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationTools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10
Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationOn one-factorizations of replacement products
Filomat 27:1 (2013), 57 63 DOI 10.2298/FIL1301057A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On one-factorizations of replacement
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationEuler Paths and Euler Circuits
Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationA characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
More informationTopological Treatment of Platonic, Archimedean, and Related Polyhedra
Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic
More informationTenacity and rupture degree of permutation graphs of complete bipartite graphs
Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationIntroduction to Graph Theory
Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationLinear Programming Problems
Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number
More informationMean Ramsey-Turán numbers
Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationLecture 4: BK inequality 27th August and 6th September, 2007
CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound
More informationUSE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS
USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This
More informationA Sublinear Bipartiteness Tester for Bounded Degree Graphs
A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationCollatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.
Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationCOUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS
COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics
More informationPermutation Betting Markets: Singleton Betting with Extra Information
Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu
More informationDefinition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
More information8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More information