Special Classes of Divisor Cordial Graphs
|
|
- Lawrence Sherman
- 7 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 varatharajansrnm@gmail.com S. Navanaeethakrishnan Department of Mathematics V.O.C. College, Tuticorin Tamil Nadu, India snk.voc@gmail.com K. Nagarajan Department of Mathematics Sri S.R.N.M.College, Sattur Tamil Nadu, India k nagarajan srnmc@yahoo.co.in 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 3-equitable 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, Addison-Wesley, 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(), 5-5. 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 ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel
More informationOn 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 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 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 informationProduct irregularity strength of certain graphs
Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer
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 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 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 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 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 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 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 informationOn Some Vertex Degree Based Graph Invariants
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan
More informationCOMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
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 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 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 informationOn the k-path cover problem for cacti
On the k-path 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 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 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 informationA NOTE ON OFF-DIAGONAL SMALL ON-LINE RAMSEY NUMBERS FOR PATHS
A NOTE ON OFF-DIAGONAL SMALL ON-LINE RAMSEY NUMBERS FOR PATHS PAWE L PRA LAT Abstract. In this note we consider the on-line Ramsey numbers R(P n, P m ) for paths. Using a high performance computing clusters,
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 informationCS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010
CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected
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 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 Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
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 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 informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS-236: 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 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 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 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, 6-70 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5
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 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 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 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 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 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 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 informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
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 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 informationFALSE ALARMS IN FAULT-TOLERANT 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 FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant
More information3. Eulerian and Hamiltonian Graphs
3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
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 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 informationOn three zero-sum Ramsey-type problems
On three zero-sum Ramsey-type 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 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 informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
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 informationGraph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
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 informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
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 informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationExtremal Wiener Index of Trees with All Degrees Odd
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of
More informationGRAPH THEORY LECTURE 4: TREES
GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection
More informationMathematics for Algorithm and System Analysis
Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationPlaying with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
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 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 informationTree-representation of set families and applications to combinatorial decompositions
Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationRising Rates in Random institute (R&I)
139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139-143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2
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 informationCacti with minimum, second-minimum, and third-minimum Kirchhoff indices
MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
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 informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationEvery Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
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 informationTriangle deletion. Ernie Croot. February 3, 2010
Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,
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 informationDegree-associated reconstruction parameters of complete multipartite graphs and their complements
Degree-associated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertex-deleted subgraphofagraphgisacard.
More informationFrans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages
Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications
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 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 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 informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
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 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 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 informationNetwork Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem
Lecture 6 Network Flow I 6. Overview In these next two lectures we are going to talk about an important algorithmic problem called the Network Flow Problem. Network flow is important because it can be
More informationDEGREE-ASSOCIATED RECONSTRUCTION PARAMETERS OF COMPLETE MULTIPARTITE GRAPHS AND THEIR COMPLEMENTS
TAIWANESE JOURNAL OF MATHEMATICS Vol. xx, No. xx, pp. 0-0, xx 20xx DOI: 10.11650/tjm.19.2015.4850 This paper is available online at http://journal.taiwanmathsoc.org.tw DEGREE-ASSOCIATED RECONSTRUCTION
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 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 informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
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 informationDiversity Coloring for Distributed Data Storage in Networks 1
Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract
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 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 informationComputer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li
Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
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 informationRECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES
RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean
More information