Special Classes of Divisor Cordial Graphs


 Lawrence Sherman
 1 years ago
 Views:
Transcription
1 International Mathematical Forum, Vol. 7,, no. 35, Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur , Tamil Nadu, India S. Navanaeethakrishnan Department of Mathematics V.O.C. College, Tuticorin Tamil Nadu, India K. Nagarajan Department of Mathematics Sri S.R.N.M.College, Sattur Tamil Nadu, India k nagarajan Abstract A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {,,..., V } such that if each edge uv is assigned the label if f(u) divides f(v) or f(v) divides f(u) and otherwise, then the number of edges labeled with and the number of edges labeled with differ by at most. If a graph has a divisor cordial labeling, then it is called divisor cordial graph. In this paper, we proved some special classes of graphs such as dragon, corona, wheel, full binary trees, G K,n and G K 3,n are divisor cordial. Mathematics Subject Classification: 5C78 Keywords: Cordial labeling, divisor cordial labeling, divisor cordial graphs Introduction By a graph, we mean a finite, undirected graph without loops and multiple edges, for terms not defined here, we refer to Harary [5].
2 738 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan First we give the some concepts in Number Theory [3]. Definition.. Let a and b be two integers. If a divides b means that there is a positive integer k such that b = ka. It is denoted by a b. If a does not divide b, then we denote a b. Graph labeling [4] is a strong communication between Number theory [3] and structure of graphs [5]. By combining the divisibility concept in Number theory and Cordial labeling concept in Graph labeling, we introduced a new concept called divisor cordial labeling [8]. In paper[8], we proved the standard graphs such as path, cycle, wheel, star and some complete bipartite graphs are divisor cordial and complete graph is not divisor cordial. In this paper we are going to prove some special classes of graphs such as full binary trees, dragon, corona, wheel, G K,n and G K 3,n are divisor cordial. A vertex labeling [4] of a graph G is an assignment f of labels to the vertices of G that induces for each edge uv a label depending on the vertex label f(u) and f(v). The two best known labeling methods are called graceful and harmonious labelings. Cordial labeling is a variation of both graceful and harmonious labelings []. Definition.. Let G = (V, E) be a graph. A mapping f : V (G) {,} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f : E(G) {,} is given by f (e)= f(u) f(v). Let v f (), v f () be the number of vertices of G having labels and respectively under f and e f (), e f () be the number of edges having labels and respectively under f. The concept of cordial labeling was introduced by Cahit []. Definition.3. A binary vertex labeling of a graph G is called a cordial labeling if v f () v f () and e f () e f (). A graph G is cordial if it admits cordial labeling. Cahit proved some results in []. Main Results Sundaram, Ponraj and Somasundaram [6] have introduced the notion of prime cordial labeling. Definition.. [6] A prime cordial labeling of a graph G with vertex set V is a bijection f from V to {,,..., V } such that if each edge uv assigned the label if gcd(f(u), f(v)) = and if gcd(f(u), f(v)) >, then the number of edges labeled with and the number of edges labeled with differ by at most.
3 Special classes of divisor cordial graphs 739 In [6], they have proved some graphs are prime cordial. Motivated by the concept of prime cordial labeling, we introduced a new special type of cordial labeling called divisor cordial labeling as follows. Definition.. [8] Let G =(V, E) be a simple graph and f : V {,,... V } be a bijection. For each edge uv, assign the label if either f(u) f(v) or f(v) f(u) and the label if f(u) f(v). f is called a divisor cordial labeling if e f () e f (). A graph with a divisor cordial labeling is called a divisor cordial graph. Example.3. Consider the following graph G F ig We see that e f () = 3 and e f () = 4. Thus e f () e f () = and hence G is divisor cordial. 3 In this paper we prove some special classes of graphs are divisor cordial. Theorem.4. Given a positive integer n, there is a divisor cordial graph G which has n vertices. Proof. Suppose n is even. Construct a path containing n + vertices v, v,..., v n + which are labeled as,,..., n + respectively. Note that the label of the edge v v is and the other edges v i v i+ ( i n + ) have the labels. Attach n vertices v n +3, v n +4,..., v n which are labeled as n +3,... n respectively, to the vertex v. We see that the labels of the edges v v i ( n + 3 i n) are. v v v v n 3 + v n + n n + v n +3 n + 3 n v n n v n Fig
4 74 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan So, we have e f () = n and e f () = + n = n and hence e f () e f (). Thus, the resultant graph G is divisor cordial., Similarly, we can construct a graph for n is odd. Theorem.5. If G is a divisor cordial graph of even size, then G e is also divisor cordial for all e E(G). Proof. Let q be the even size of the divisor cordial graph G. Then it follows that e f () = e f () = q. Let e be any edge in G which is labeled either or. Then in G e, we have either e f () = e f () + or e f () = e f () + and hence e f () e f (). Thus G e is divisor cordial. Theorem.6. If G is a divisor cordial graph of odd size, then G e is also divisor cordial for some e E(G). Proof. Let q be the odd size of the divisor cordial graph G. Then it follows that either e f () = e f () + or e f () = e f () +. If e f () = e f () + then remove the edge e which is labeled as and if e f () = e f () + then remove the edge e which is labeled from G. Then it follows that e f () = e f (). Thus G e is divisor cordial for some e E(G). Note.7. Similarly we can prove for the divisor cordiality of G + e. Definition.8. An ordered rooted tree is a binary tree if each vertex has atmost two children. Definition.9. A full binary tree is a binary tree inwhich each internal vertex has exactly two children. Next we will prove that a full binary tree is divisor cordial. Theorem.. Every full binary tree is divisor cordial. Proof. We note that every full binary tree has odd number of vertices and hence has even number of edges. Let T be a full binary tree and let v be a root of T which is called zero level vertex. Clearly, the i th level of T has i vertices. If T has m levels, then the number of vertices of T is m+ and the number of edges is m+. Now assign the label to the root v and assign the labels 3 and to the first level vertices. Next, we assign the labels i, i +,..., i+ to the i th level vertices for i m In zeroth, first and second levels, we have e f () = e f () = 3. Clearly, i + j i+ + j for j i and i + j i+ + j for j i. Then after the second levels, we have e f () = e f () = m 3. Thus e f () e f () = and hence T is divisor cordial.
5 Special classes of divisor cordial graphs 74 The next example shows that the divisor cordiality of full binary tree upto 4 levels. Example.. Here we see that e f () = e f () = F ig 3 Theorem.. Let G be any divisor cordial graph of size m and K,n be a bipartite graph with the bipartition V = V V with V = {x, x } and V = {y, y,... y n }. Then the graph G K,n obtained by identifying the vertices x and x of K,n with that labeled and the largest prime number p such that p m respectively in G is also divisor cordial. Proof. Let G be any divisor cordial graph of size m. Let v k and v l be the vertices having the labels and the largest prime number namely p such that p m. Let V = V V be the bipartition of K,n such that V = {x, x } and V = {y, y,... y n }. Now assign the labels m +, m +,... m + n to the vertices y, y,..., y n respectively. Now identify the vertices x and x of K,n with that labeled and the largest prime number p such that p m respectively in G. Case (i) p < m + n < p Then the multiples of p are not available in the labels of y i ( i n). Then we see that the edges of K,n incident with the vertex v k have the label and with the vertex v l have the label. Thus the edges of K,n contribute equal numbers namely n to both e f () and e f () in G K,n. Hence G K,n is divisor cordial. Case (ii) m + n p.
6 74 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan Let q be the largest prime number such that q m + n which is labeled to some y i. Then interchange the labels of v l and y i, that is p and q. We observe that the largest prime number q does not divide the labels of y, y,..., y n. So, again the edges of K,n incident with the vertex v l have the labels and hence G K,n is divisor cordial. The following example explains the construction developed in the Theorem.. Example.3. Consider the following graph G Fig 4 5 Here e f () = 4 and e f () = 3 and hence e f () e f () =. Thus G is divisor cordial. Case(i): Now consider the bipartite graph K,3. Then the graph G K,3 and its labels are gives as follows.
7 Special classes of divisor cordial graphs Fig 5 Here e f () = 7 and e f () = 6 and e f () e f () =. Thus G K,3 is divisor cordial. Case (ii): Now consider the bipartite graph K,5. The graph G K,5 and its labels are given as follows.
8 744 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan Fig 6 Here we interchanged the labels 5 and and e f () = 9 and e f () =8 and so e f () e f () = Thus G K,5 is divisor cordial. Theorem.4. Let G be any divisor cordial graph of size m and K 3,n be a bipartite graph with the bipartition V = V V with V = {x, x, x 3 } and V = {y, y,... y n }, where n is even. Then the graph G K 3,n obtained by identifying the vertices x, x and x 3 of K 3,n with that labeled, labeled and the largest prime number p such that p m respectively in G is also divisor cordial. Proof. Let G be any divisor cordial graph of size m. Then e f () e f (). Let v k, v l and v r be the vertices having the labels, and the largest prime number namely p such that p m. Let V = V V be the bipartition of K 3,n such that V = {x, x, x 3 } and V = {y, y,... y n }. Now assign the labels m +, m +,... m + n to the vertices y, y,..., y n respectively. Now identify the vertices x, x and x 3 of K 3,n with that labeled, labeled and and the largest prime number p such that p m respectively in G. Case (i) p < m + n < p Then the multiples of p are not available in the labels of y i ( i n). Then we see that the edges of K 3,n incident with the vertex v k have the label and with the vertex v r have the label. Since n divides only even numbers, n edges of K 3,n incident with the vertex v l have the label and remaining n edges have the label. Thus the edges of K 3,n contribute equal numbers
9 Special classes of divisor cordial graphs 745 namely 3n to both e f () and e f () in G K 3,n. Hence G K 3,n is divisor cordial. Case (ii) m + n p. Let q be the largest prime number such that q m + n which is labeled to some y i. Then interchange the labels of v r and y i, that is p and q. We observe that the largest prime number q does not divide the labels of y, y,..., y n. So, again the edges of K 3,n incident with the vertex v r have the labels and hence G K 3,n is divisor cordial. Note.5. Theorem.4 is also valid for m is even and n is odd. If both m and n are odd, the above labeling pattern is not valid. The following example illustrates the construction developed in the Theorem.4. Example.6. Consider the following divisor cordial graph G Fig 7 The divisor cordiality of G K 3,4 is shown below.
10 746 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan Fig 8 Here e f () = e f () =. Now we present divisor cordial labeling for the graphs obtained by joining apex vertices of two stars to a new vertex. We extend this result for three copies of stars. Definition.7. Consider two stars K (),n and K (),n. Then G =< K (),n, K (),n > is the graph obtained by joining apex (central) vertices of stars to a new vertex x. Note that G has n + 3 vertices and n + edges. Definition.8. Consider t copies of stars namely K,n, () K (),n,..., K (t),n. Then G =< K (),n, K (),n,..., K (t),n > is the graph obtained by joining apex vertices of each K (m ),n and K (m),n to a new vertex x m where m t. Note that G has t(n + ) vertices and t(n + ) edges. First we will prove that < K (),n, K (),n > is divisor cordial. Theorem.9. The graph G =< K (),n, K (),n > is divisor cordial. Proof. Let v (), v (),..., v n () be the pendant vertices of K (),n and v (), v (),..., v n () be the pendant vertices of K (),n. Let c and c be the apex
11 Special classes of divisor cordial graphs 747 vertices of K (),n and K (),n respectively and they are adjacent to a new common vertex x. Now assign the label to c and the largest prime number p such that p n+3 to c and the remaining labels to the vertices of G. Since divides any integer the edges incident to c contribute n + to e f () and since p does not divide any labels of the vertices adjacent to c, the edges incident to c also contribute n + to e f (). Hence e f () e f () =. Thus G is divisor cordial. Next we will extend this result to 3 stars as follows. Theorem.. The graph G =< K (),n, K (),n, K (3),n > is divisor cordial. Proof. Let Let v (i), v (i),..., v n (i) be the pendant vertices of K (i),n and let c i be the apex vertex of K (i),n for i =,, 3. Now c and c are adjacent to x and c and c 3 are adjacent to x. Note that G has 3n + 5 vertices and 3n + 4 edges. Now assign the label to c, to c and p to c 3 where p is the largest prime number such that p 3n + 5. Then assign the odd and even labels equally to the vertices v (), v (),..., v n (), x, x if n is even. Suppose n is odd, then assign n+ odd labels and n+3 even labels to these vertices. Then assign remaining labels to the remaining vertices. Since divides any integer, the edges incident to c contribute n + to e f (). Since p does not divide any labels of the vertices adjacent to c 3, the edges incident to c 3 also contribute n + to e f (). Since divides even labels and does not divide odd labels, the edges incident to c contribute n+ ( n+) to e f() and n+ ( n+3) to e f() if n is even (odd). Thus, if n is even, then e f () = e f () = n + + n+ = 3n+4 and if n is odd then e f () = n + + n+ =n+ and e f () = n + + n+3 = n + and hence e f () e f (). Thus G is divisor cordial. The Theorems.9 and. are explained using the following example. Example.. Consider the graph G =< K (),8, K (),8 >.
12 748 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan x Fig 9 Here e f () = e f () = 9. Example.. Consider the graph G =< K (),7, K (),7, K (3),7 > x x Fig 7 9 Here e f () = and e f () = 3 and e f () e f () =. References [] I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars combinatoria, 3(987), 7. [] I. Cahit, On cordial and 3equitable labelings of graph, Utilitas Math, 37(99), [3] David M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown Company Publishers, 98.
13 Special classes of divisor cordial graphs 749 [4] J. A. Gallian, A dynamic survey of graph labeling, Electronic Journal of Combinatorics, 6(9), DS6. [5] F. Harary, Graph Theory, AddisonWesley, Reading, Mass, 97. [6] M. Sundaram, R. Ponraj and S. Somasundaram, Prime cordial labeling of graphs, Journal of Indian Academy of Mathematics, 7(5) [7] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, Cordial and 3 equitable labeling for some star related graphs, International mathematical Forum, 4, 9, no. 3, [8] R. Varatharajan, S. Navaneethakrishnan and K. Nagarajan, Divisor cordial labeling of graphs, International Journal of Mathematical Combinatorics, Vol. 4(), 55. Received: November,
P. Jeyanthi and N. Angel Benseera
Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONEPOINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel
More informationSome Polygonal Sum Labeling of Paths
Volume 6 No.5, January 03 Some Polygonal Sum Labeling of Paths S.Murugesan Associate Professor CBM college Coimbatore60 D.Jayaraman Associate Professor CBM College Coimbatore 60 J.Shiama Assistant Professor
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationPath Relatively Prime Cordial Graph
World Journal of Research and Review (WJRR) ISSN:24553956, Volume3, Issue2, August 2016 Pages 7477 Path Relatively Prime Cordial Graph Dr. A. Nellai Murugan,V.Nishanthini Abstract be a graph with vertices
More informationHomework 15 Solutions
PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition
More informationON SUPER (a, d)edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS
Discussiones Mathematicae Graph Theory 3 (01) 535 543 doi:10.7151/dmgt.163 ON SUPER (a, d)edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS P. Roushini Leely Pushpam Department of Mathematics
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationCharacterizations of Arboricity of Graphs
Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs
More informationSolutions to Exercises 8
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.
More informationDifference Labeling of Some Graph Families
International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 4767 PISSN: 23214759 Volume 2 Issue 6 June. 2014 PP3743 Difference Labeling of Some Graph Families Dr. K.Vaithilingam
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 informationDistance Two VertexMagic Graphs
Distance Two VerteMagic Graphs Ebrahim Salehi Department of Mathematical Sciences Universit of Nevada Las Vegas Las Vegas, NV 89544020 ebrahim.salehi@unlv.edu Abstract Given an abelian group A, a graph
More informationChapter 2. Basic Terminology and Preliminaries
Chapter 2 Basic Terminology and Preliminaries 6 Chapter 2. Basic Terminology and Preliminaries 7 2.1 Introduction This chapter is intended to provide all the fundamental terminology and notations which
More informationZeroSum Magic Labelings and Null Sets of Regular Graphs
ZeroSum Magic Labelings and Null Sets of Regular Graphs Saieed Akbari a,c Farhad Rahmati b Sanaz Zare b,c a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department
More informationTree Cover of Graphs
Applied Mathematical Sciences, Vol. 8, 2014, no. 150, 74697473 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.49728 Tree Cover of Graphs Rosalio G. Artes, Jr. and Rene D. Dignos Department
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 CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
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 informationProduct irregularity strength of certain graphs
Also available at http://amc.imfm.si ISSN 18553966 (printed edn.), ISSN 18553974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationSample Problems in Discrete Mathematics
Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a prerequisite to Computer Algorithms Try to solve all of them You should also read
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationOn a tree graph dened by a set of cycles
Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor NeumannLara b, Eduardo RiveraCampo c;1 a Center for Combinatorics,
More informationGraph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.
Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.
More informationGraph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania
Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics
More informationDivisor graphs have arbitrary order and size
Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationA CHARACTERIZATION OF TREE TYPE
A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying
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 informationTools for parsimonious edgecolouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR201010
Tools for parsimonious edgecolouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR201010 Tools for parsimonious edgecolouring of graphs
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 informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationCMSC 451: Graph Properties, DFS, BFS, etc.
CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
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 informationGraph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick
Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements
More informationPrinciple of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))
Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction
More informationOn the kpath cover problem for cacti
On the kpath cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
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 informationOn Some Vertex Degree Based Graph Invariants
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723730 ISSN 03406253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
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 informationCSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24
CSL851: Algorithmic Graph Theory Semester I 20132014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
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 information7. Colourings. 7.1 Vertex colouring
7. Colourings Colouring is one of the important branches of graph theory and has attracted the attention of almost all graph theorists, mainly because of the four colour theorem, the details of which can
More informationDiscrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 Email: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
More informationOn Total Domination in Graphs
University of Houston  Downtown Senior Project  Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper
More informationCSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph
Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian 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 informationNotes on Matrix Multiplication and the Transitive Closure
ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.
More informationTopological Entropy of Golden Mean Lookalike Shift Spaces
International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 4767 PISSN: 23214759 Volume 2 Issue 7 July 2014 PP0510 Topological Entropy of Golden Mean Lookalike Shift Spaces Khundrakpam
More informationGeneralizing the Ramsey Problem through Diameter
Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive
More information(Vertex) Colorings. We can properly color W 6 with. colors and no fewer. Of interest: What is the fewest colors necessary to properly color G?
Vertex Coloring 2.1 33 (Vertex) Colorings Definition: A coloring of a graph G is a labeling of the vertices of G with colors. [Technically, it is a function f : V (G) {1, 2,...,l}.] Definition: A proper
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationSimple Graphs Degrees, Isomorphism, Paths
Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week MultiGraph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
More informationA simple proof for the number of tilings of quartered Aztec diamonds
A simple proof for the number of tilings of quartered Aztec diamonds arxiv:1309.6720v1 [math.co] 26 Sep 2013 Tri Lai Department of Mathematics Indiana University Bloomington, IN 47405 tmlai@indiana.edu
More informationThe origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)
Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph
More informationA graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices.
1 Trees A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices. Theorem 1 For every simple graph G with n 1 vertices, the following
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationHOMEWORK #3 SOLUTIONS  MATH 3260
HOMEWORK #3 SOLUTIONS  MATH 3260 ASSIGNED: FEBRUARY 26, 2003 DUE: MARCH 12, 2003 AT 2:30PM (1) Show either that each of the following graphs are planar by drawing them in a way that the vertices do not
More informationA NOTE ON OFFDIAGONAL SMALL ONLINE RAMSEY NUMBERS FOR PATHS
A NOTE ON OFFDIAGONAL SMALL ONLINE RAMSEY NUMBERS FOR PATHS PAWE L PRA LAT Abstract. In this note we consider the online Ramsey numbers R(P n, P m ) for paths. Using a high performance computing clusters,
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 informationMath 443/543 Graph Theory Notes 4: Connector Problems
Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we
More informationSEMITOTAL AND TOTAL BLOCKCUTVERTEX GRAPH
CHAPTER 3 SEMITOTAL AND TOTAL BLOCKCUTVERTEX 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 Bradius
More informationOn the Ramsey Number R(4, 6)
On the Ramsey Number R(4, 6) Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 ge@cs.indstate.edu Submitted: Mar 6, 2012; Accepted: Mar 23, 2012;
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 informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationA MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY
International Journal of Pure and Applied Mathematics Volume 03 No. 05, 670 ISSN: 38080 (printed version); ISSN: 343395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5
More informationCS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010
CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison GomoryHu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected
More informationON THE TREE GRAPH OF A CONNECTED GRAPH
Discussiones Mathematicae Graph Theory 28 (2008 ) 501 510 ON THE TREE GRAPH OF A CONNECTED GRAPH Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria,
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 informationCOLORED GRAPHS AND THEIR PROPERTIES
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring
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 informationA note on properties for a complementary graph and its tree graph
A note on properties for a complementary graph and its tree graph Abulimiti Yiming Department of Mathematics Institute of Mathematics & Informations Xinjiang Normal University China Masami Yasuda Department
More informationp 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)
.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other
More informationDISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS
1 2 Discussiones Mathematicae Graph Theory xx (xxxx) 1 9 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 DISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS Christian Barrientos Department of Mathematics,
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
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
More informationA counterexample to a result on the tree graph of a graph
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 368 373 A counterexample to a result on the tree graph of a graph Ana Paulina Figueroa Departamento de Matemáticas Instituto Tecnológico
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationIntersection Dimension and Maximum Degree
Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai  600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show
More informationOutline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1
GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept
More informationCOMBINATORIAL CHESSBOARD REARRANGEMENTS
COMBINATORIAL CHESSBOARD REARRANGEMENTS DARYL DEFORD Abstract. Many problems concerning tilings of rectangular boards are of significant combinatorial interest. In this paper we introduce a similar type
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationPlanar Graph and Trees
Dr. Nahid Sultana December 16, 2012 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem Tree Spanning Trees Minimum Spanning
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 informationFALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem
Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULTTOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of faulttolerant
More informationOn three zerosum Ramseytype problems
On three zerosum Ramseytype problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics
More information1 Basic Definitions and Concepts in Graph Theory
CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered
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 informationPlanarity Planarity
Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?
More informationDiscrete Applied Mathematics. The firefighter problem with more than one firefighter on trees
Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with
More informationTheorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.
Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationBasic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by
Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
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 HongJian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,
More informationConstructing Zero Divisor Graphs
Constructing Zero Divisor Graphs Alaina Wickboldt, Louisiana State University Alonza Terry, Xavier University of Louisiana Carlos Lopez, Mississippi State University SMILE 2011 Outline Introduction/Background
More informationarxiv: v2 [math.co] 30 Nov 2015
PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out
More information