MAD 3105 PRACTICE TEST 2 SOLUTIONS

Save this PDF as:

Size: px
Start display at page:

Transcription

1 MAD 3105 PRACTICE TEST 2 SOLUTIONS 1. Define a graph G with V (G) = {a, b, c, d, e}, E(G) = {r, s, t, u, v, w, x, y, z} and γ, the function defining the edges, is given by the table ɛ r s t u v w x y z γ(ɛ) (a, b) (a, d) (d, a) (a, b) (c, d) (d, d) (e, e) (e, a) (e, e) (a) Draw a picture of G. x e z a y s t d w r u v b c (b) directed (c) deg(b) = 2 and deg(e) = 5 2. List 5 properties that are invariant under isomorphism. See section 10 in the lecture notes for Introduction to Graphs. 3. Sketch a graph of each of the following when n = 5. For what positive value(s) of n > 2 is the graph bipartite? Look in the text in section 8.2 for graphs. (a) K n Bipartite if n = 2 only (b) C n Bipartite if n is even and at least 3. (c) W n Never Bipartite (d) Q n Bipartite for all n > Draw all the nonisomorphic simple graphs with 6 vertices and 4 edges. 1

2 MAD 3105 PRACTICE TEST 2 SOLUTIONS 2 5. Determine which of the following graphs are isomorphic. 1 2 a b 5 6 e f 3 4 c d G 1 G 2 are isomorphic. None of the others are isomorphic to another. 6. Given the graph G below, how many different isomorphisms are there from G to G. Briefly explain. There are 8 different isomorphisms: d->d e->e d->e e->d c->c b->b a->a c->b b->c a->a c->g b->f a->h c->f b->g a->h f->f g->g h->h f->g g->f h->h f->f g->g h->h f->g g->f h->h f->b g->c h->a f->c g->b h->a f->b g->c h->a f->c g->b h->a 7. Can an undirected graph have 5 vertices, each with degree 6? Yes, Take for example the complete graph with 5 vertices and add a loop at each vertex. 8. Can a simple graph have 5 vertices, each with degree 6? No, the complete graph with 5 vertices has 10 edges and the complete graph has the largest number of edges possible in a simple graph.

3 MAD 3105 PRACTICE TEST 2 SOLUTIONS 3 9. A graph has 21 edges has 7 vertices of degree 1, three of degree 2, seven of degree 3, and the rest of degree 4. How many vertices does it have? The graph has 19 vertices. 10. How many edges does a graph with 5 vertices have if 2 of the vertices have degree 3, 1 vertex has degree 2, and the rest of the vertices have degree 1? By the Hand Shaking Theorem, 2 E = = 10, so E = Let S be a set of simple graphs and define the relation R on S as follows: Let G, H S. Then (G, H) R if and only if G H. This is equivalent to Let G, H S. Then (G, H) R if and only if there exists an isomorphism f : V (G) V (H). Prove this relation is an equivalence relation. This is the proof of Theorem 4 in the lecture notes on Equivalence Relations. 12. Suppose G and H are isomorphic simple graphs. Show that their complimentary graphs G and H are also isomorphic. Proof. Since G and H are isomorphic there is a function f : V (G) V (H) that is a bijection and preserves adjacency. Note that a complimentary graph has the exact same vertex set, so f may also be thought of as a function from V (G) to V (H). Moreover, f is still a bijection. The question that remains is does f preserve adjacency in the complimentary graphs? Let u, v V (G) = V (G). {u, v} is an edge in G iff {u, v} is not an edge in G by the definition of complimentary graph. {u, v} is not an edge in G iff {f(u), f(v)} is not an edge in H since f preserves adjacency. {f(u), f(v)} is not an edge in H iff {f(u), f(v)} is an edge in H by the definition of complimentary graphs. Thus {u, v} is an edge in G iff {f(u), f(v)} is an edge in H. This shows f preserves adjacency. 13. Give the number of cut vertices and cut edges of the following graphs. (a) K n, n 2 For n = 2 there is one cut edge. For n 3 there are no cut edges. For n 2 there are no cut vertices. (b) W n, n 3 There are no cut edges nor vertices. (c) K m,n, m, n 1 If m = n = 1 there is one cut edge and no cut vertices. If m = 1 and n > 1 there are n cut edges and one cut vertex. If n = 1 and m > 1 there are m cut edges and one cut vertex. If m, n 2 there are no cut edges nor vertices. 14. For what values of n does each graph have (i) and Euler circuit? (ii) a Hamilton circuit? (a) K n

4 MAD 3105 PRACTICE TEST 2 SOLUTIONS 4 (i) n odd and at least 3 (ii) all n (b) C n (i) and (ii) All n 3 (c) W n (i) no n (ii) All n 3 (d) Q n (i) n even and at least 2 (ii) all n. 15. Does the Theorem given imply the graph below has a Hamilton circuit? The given theorem does not imply anything about the graph. The condition of the theorem is not satisfied. 16. Draw all the nonisomorphic (unrooted) trees with 6 edges. 17. Draw all the nonisomorphic rooted trees with 4 edges.

5 MAD 3105 PRACTICE TEST 2 SOLUTIONS Given G is a finite simple graph. Give six different completions to the sentence. G is a tree if and only if... Look in the lecture notes Introduction to Trees pages 1 (definition, Theorem 1) and 7 (Theorem 5). 19. Answer the following questions. Explain. (a) How many leaves does a full 3-ary tree with all leaves at height 4 have? L = 3 height = 3 4 = 81 (b) How many leaves does a 3-ary tree have if it has 15 parents and every parent has exactly 3 children? There are = 46 vertices since every vertex except the root is the child of one of these 15 parents. Thus the number of leaves is = Prove (a) Prove that every connected graph with at least 2 vertices has at least two non-cut vertices. Text section 8.4, problem 29. (b) Prove a connected graph with n vertices has at least n 1 edges. One version uses the first principal of induction and problem 20a. The following is another possible version. Proof. Basis: A graph with a single vertex clearly has at least 0 edges. Induction Step: Let n be a positive integer and suppose any connected graph with k vertices has at least k 1 edges for any integer k with 1 k n. Let G be a connected graph with n + 1 vertices and let v be a vertex of G. Remove v and all incident edges to v from G. The resulting graph, which we will call G, may or may not be connected, but we do know it has n vertices. Suppose G has s components where 1 s n. Each

6 MAD 3105 PRACTICE TEST 2 SOLUTIONS 6 component is a connected graph with n or fewer vertices, so we may apply the induction hypothesis to each component. For each component, if the component has k vertices then it has at least k 1 edges. Thus G must have at least n s edges. When v was removed from G we only removed edges incident to v to obtain G. For each component in G there must be an edge from v to a vertex in that component since G was connected. Thus there must be at least s more edges in G than there are in G, i.e. the number of edges in G is at least (n s) + s = n. (c) Proof. Let G be a finite graph with all vertices of degree at least 2. We prove this theorem by recursively creating a path in G that must eventually contain a cycle. Basis: Choose an arbitrary vertex, v 0, in G. Since deg(v 0 ) 2 there is an edge, {v 0, v 1 }, incident to v 0. Take this edge as the first edge in the path. Recursive Step: Suppose we have a path P n : {v 0, v 1 }, {v 1, v 2 },..., {v n 1, v n } in G. If v n is not equal to any of the vertices v 0, v 1,..., v n 1, then the only edge in P n incident to v n is {v n 1, v n }. Since deg(v n ) 2, there must be an edge {v n, v n+1 } in G different from any of the edges in the path P n. Add this edge onto the end of the path P n to obtain a path P n+1. If v n is equal to one of the vertices v 0, v 1,..., v n 1, then we stop with P n. Notice the recursive step must be possible until the ending condition is met and the ending condition must eventually be met since there are only finitely many vertices in G. The resulting path must contain a cycle since the path repeats a vertex. The cycle would go from the first occurrence of the vertex to the second occurrence. (d) Prove a graph with n vertices and at least n edges contains a cycle for all positive integers n. You may use that a graph with all vertices of degree at least 2 contains a cycle. Proof. Basis: Prove a graph with 1 vertex and at least 1 edge contains a cycle. Notice that is there is only one vertex, then any edge must begin and end at that vertex. Thus, in this case the graph contains a loop, which is a cycle. Induction Step: Let k Z +. Assume that any graph with k vertices and at least k edges contains a cycle. Prove any graph with k + 1 vertices and at least k + 1 edges contains a cycle. Let G be a graph with k + 1 vertices and at least k + 1 edges. We break the rest of the proof into 2 cases. Case 1: Assume all the vertices have degree at least 2. Problem 20c tells us in this case, that there must be a cycle contained in G.

7 MAD 3105 PRACTICE TEST 2 SOLUTIONS 7 Think about why you cannot use something similar to what we do below in this case. Case 2: Assume there is at least one vertex of degree 0 or 1. It is not necessarily true that there is a vertex with degree 0 AND one with degree 1. Case A: Suppose there exists some vertex, say v, in G with degree 0. This vertex must be isolated. In other words, there are no edges attached to v. Consider The graph G = G {v}. G has k vertices and at least k edges. It actually has at least k + 1 edges, but that s ok. We can apply the induction hypothesis to G to get that G must contain a cycle. However, G is a subgraph of G, so a cycle in G is also a cycle in G. Case B: Suppose there exists some vertex, say v, in G with degree 1. This vertex must have exactly one edge, say e, attached to it. Let G be the subgraph of G obtained by removing v and e. G has k vertices and at least k edges. We can apply the induction hypothesis to G to get that G must contain a cycle. However, G is a subgraph of G, so a cycle in G is also a cycle in G. We have shown in all possible cases G must contain a cycle, therefore, by the principle of mathematical induction, any graph with at least as many edges as vertices must contain a cycle. (e) Prove the complimentary graph of a disconnected graph is connected. Proof. Let G be a graph that is disconnected and let u and v be vertices in the complimentary graph, which we will denote by G. Recall that the vertex set of G and G are the same so u and v are also vertices in G. We break the proof into two cases. First the case where u and v are not in the same component of G and second the case where they are in the same component. Case 1: Assume u and v are not in the same component in G. Then {u, v} is not and edge in G. By the definition of the complimentary graph, {u, v} must be an edge in G. This edge provides a path in G from u to v. Case 2: Assume u and v are in the same component in G. There there is a vertex w in G that is a different component than the one u and v are in. Since it is in a different component, {u, w}, {w, v} are not edges in G, so {u, w}, {w, v} are edges in G. These edges give us a path from u to v in G. This shows G is connected. (f) A full m-ary tree with i internal vertices has n = mi + 1 vertices Lecture notes Introduction to Trees, Theorem 3 part 1. (g) An m-ary tree with height h has at most m h leaves. Lecture notes Introduction to Trees, Theorem 4. (h) Let G be a simple graph. G is a tree if and only if G is acyclic but the addition of any edge between any two vertices in G will create a cycle in G. Lecture notes Introduction to Trees, Theorem 5.

8 MAD 3105 PRACTICE TEST 2 SOLUTIONS 8 (i) Let G be a simple graph. G is a tree if and only if G is connected but the removal of any edge in G produces a disconnected graph. Proof. Suppose G is a tree. Then by defintion, G is connected. Let {u, v} be an arbitrary edge in G. Then {u, v} is clearly a path from u to v. Since G is a tree this is the only path in G between u and v, so if the edge {u, v} is removed from G there will be no path from u to v in the resulting graph. Thus the resulting graph is disconnected. Conversely, suppose G is connected but the removal of any edge in G produces a disconnected graph. We need to prove G is a tree. We already have G is connected, so what we still need to show is that G is acyclic. We use contradiction. Suppose G contains a cycle, C, and {u, v} is an edge in C. If we remove this edge we claim the resulting graph, say G, is still connected. Let w and x be vertices in G. Since G is connected there is a path from w to x in G. If this path uses the edge {u, v}, then we can replace this edge in the path with the part of the cycle left from C after removing {u, v}. The new path is a path from w to x that does not use {u, v}, so it is a path in G. If the original path does not use {u, v} then it is a path in G. Thus there is a path in G from w to x. since these vertices were arbitrary we have shown G is connected. This contradicts our assumption that the removal of any edge in G produces a disconnected graph. Thus G does not contain a cycle and must, therefore, be a tree. (j) Let G be a simple graph with n vertices. G is a tree is and only if G is connected and has n 1 edges. Text p. 642, exercise 15. (k) Let G be a simple graph with n vertices. G is a tree if and only if G is acyclic and has n 1 edges. Proof. Let G be a simple graph with n vertices. Assume G is a tree. Then by definition G is acyclic. Select a root v and direct the edges away from v. Then there are as many edges as there are terminal vertices of the edges and every vertex except v is a terminal vertex of some edge. Conversely, assume G is acyclic and has n 1 edges and we will prove G is a tree. To show G is a tree we only need to show it is connected. Suppose G has k connected components. Then the vertices and edges of G must be distributed among the components. Each component must be a tree since the graph is acyclic. Since each component is a tree each component must have exactly one less edge than vertices in that component. But then the total number of edges in G would be n k. Hence k = 1 and so G must be connected.

9 MAD 3105 PRACTICE TEST 2 SOLUTIONS Find the prefix, postfix, and infix forms for the algebraic expressions ab + c/d 3 and a((b + c)/d) 3 (in infix form). prefix: + a b / c d 3; a / + b c d 3 postfix: a b c d 3 / +; a b c + d / What is the value of the prefix expression: / = What is the value of the postfix expression: 9 3 / = 40

GRAPH 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

Outline 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

Discrete 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

Euler 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

3. 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,

Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

Simple 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

Full 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

Divisor 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

Analysis 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

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

Handout #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

Graph 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

Midterm 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

Discrete 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...............................

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

1 Definitions. Supplementary Material for: Digraphs. Concept graphs

Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.

Class 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

Cycles 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

A 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

SHARP 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.,

I. 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

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

Introduction 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

Section Summary. Introduction to Graphs Graph Taxonomy Graph Models

Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems

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

Foundations of Geometry 1: Points, Lines, Segments, Angles

Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.

CS 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

Mathematical 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

Degree 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

Zachary 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

3. 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

SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

Total 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

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

. 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

Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

6.2 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

Connectivity 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

3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

An 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

Examination 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

BOUNDARY 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

Single 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

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

Graphs 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

Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

Product 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

Lecture 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

USA Mathematical Talent Search

ID#: 678 7 We shall approach the problem as follows: first, we will define a function and some of its properties; then, we will solve the problem using the function we have defined. Let f y (x) have domain

Handout #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

6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

The Independence Number in Graphs of Maximum Degree Three

The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684

All 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

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

The 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:

WOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology

First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121

COMBINATORIAL 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

C H A P T E R Regular Expressions regular expression

7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

136 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

Just 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

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie. An Iterative Compression Algorithm for Vertex Cover

Friedrich-Schiller-Universität Jena Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie Studienarbeit An Iterative Compression Algorithm for Vertex Cover von Thomas Peiselt

Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

6.852: Distributed Algorithms Fall, 2009. Class 2

.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

SCORE 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

Labeling 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

Mathematical 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

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

15-750 Graduate Algorithms Spring 2010 1 Cheap Cut (25 pts.) Suppose we are given a directed network G = (V, E) with a root node r and a set of terminal nodes T V. We would like to remove the fewest number

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

MATH 131 SOLUTION SET, WEEK 12

MATH 131 SOLUTION SET, WEEK 12 ARPON RAKSIT AND ALEKSANDAR MAKELOV 1. Normalisers We first claim H N G (H). Let h H. Since H is a subgroup, for all k H we have hkh 1 H and h 1 kh H. Since h(h 1 kh)h 1

On 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

8. 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,

Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

An algorithmic classification of open surfaces

An algorithmic classification of open surfaces Sylvain Maillot January 8, 2013 Abstract We propose a formulation for the homeomorphism problem for open n-dimensional manifolds and use the Kerekjarto classification

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

8.1 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

Network 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

Tree-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

Data Structures and Algorithms Written Examination

Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where

Math 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

Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS

ZERO-DIVISOR AND IDEAL-DIVISOR GRAPHS OF COMMUTATIVE RINGS T. BRAND, M. JAMESON, M. MCGOWAN, AND J.D. MCKEEL Abstract. For a commutative ring R, we can form the zero-divisor graph Γ(R) or the ideal-divisor

Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

An 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,