MT365 Examination 2017 Part 1 Solutions Part 1

Size: px
Start display at page:

Download "MT365 Examination 2017 Part 1 Solutions Part 1"

Transcription

1 MT xamination 0 Part Solutions Part Q. G (a) Number of vertices in G =. Number of edges in G = (i) The graph G is simple no loops or multiple edges (ii) The graph G is not regular it has vertices of deg., and vertices of deg. (iii) The graph G is bipartite the vertices can be split into two sets (circled and uncircled above) so that edges only join vertices in one set to those in the other. Q. S,9, T,,,,, S,,, 0,,,9 T S,, T (a) Q The network N changed to a basic network with flows, capacities and saturated arcs low ugmenting Path found alphabetically is SS T T. It will carry a flow of and will saturate. (Note the labels found are not asked for) G Noel astham 0

2 MT xamination 0 Part Solutions Part Q 9 (a) T is bicentral. Vertices removed first {,,,,, 9} second {, } icentre and Remove Sequence (, Remove Sequence (,, Remove Sequence (,,, Remove Sequence (,,,, Remove Sequence (,,,,, Remove Sequence (,,,,,, Remove Sequence (,,,,,,, Remove 9 Sequence (,,,,,,, ) inal Prufer Sequence (,,,,,,, ) Q S 0 T rcs and are redundant and so have been omitted Noel astham 0

3 MT xamination 0 Part Solutions Part Q. (a) c r c r c r r r c c c Note that: vertices and edges does not necessarily mean we have a tree! Since we have a cycle! r c c The framework is not rigid the section c r c r c is disconnected from the rest of the bipartite graph as indicated by the heavy lines in red above. (c) One bracing when added which will make the framework rigid is r c. Q. ( The graph is planar since f = and v e + f = + = ) (a) The face degrees are :- face of degree and 9 faces of degree. The sum of the face degrees = deg f = x + 9 x = + = Twice the number of edges = e = x = deg f = e as required G or the graph G there is a four colouring of the edges as above so χ (G) y Vizing's Theorem χ (G) since the maximum vertex degree is Hence χ (G) = Noel astham 0

4 MT xamination 0 Part Solutions Part Q (a) w x y z w x y 0 0 z n optimum assignment is w, z, y, x. Q9 The given codewords are 0, 0, 0 (a) The remaining five coewords are 0000, 0, 00,, cycles to itself cycles to itself cycles 0 cycles The code is thus cyclic. (c) 0 0 G= thus H= [ ] Q M T H S R (a) H M S T R (i) S:- HMTS (ii) S:- HMST Q (a) v = v a, v = v b + v c, v = -v c i = i a, i = i b - i c, (i = i b since i i = i b - i a ) Noel astham 0

5 MT xamination 0 Part Part Solutions Q (a) (i) = (ii) Out-degree sequence : (,,,, ) In-degree sequence : (,,,, ) (,) (,) (,) (,) (In,Out) (,) (i) cycle of length starting at vertex is (ii) closed trail in of length from vertex is (iii) closed walk in of length from vertex is (iv) walk in of length from vertex to vertex is (c) (i) = = There are walks from vertex to vertex in. (ii) The walks of length from vertex to vertex are and (d) Note as given in the question has element a incorrect but this does not affect the answer = = 0 y Theorem. of Networks :- If = has all non-diagonal entries > 0 then the digraph is strongly connected, but + + has all non-diagonal entries > 0 except for the entry in position (,) and so will have the same. We note that there is a walk of length from vertex to vertex as in (iv) above. So is strongly connected. (e) (i) (ii) very strongly connected digraph is Hamiltonian is false. igraph above is a counterexample since it is strongly connected but there is no Hamiltonian cycle as we must go from to and each path out of omits a vertex before retuning to. very Hamiltonian digraph is strongly connected is true since the Hamiltonian cycle provides a path between each pair of vertices. MT xam 0 Solutions Page of

6 MT xamination 0 Part Part Solutions Q (a) hoose of length the shortest length hoose of length next shortest length hoose of length next shortest length hoose of length 9 next shortest length an't choose (), (), (), (), () all create cycles hoose of length next shortest length not creating a cycle Length of the minimum connector =. Remove vertex and start with vertex hoose edge of length smallest from hoose edge of length smallest from, hoose edge of length smallest from,, hoose edge of length smallest from,,, 9 Lower bound for TSP = + +! = + = (c) Remove vertex Lower bound for TSP = = 0 + = 9 0 (d) The better lower bound is since it is the larger of the two (e) If we remove S in calculating a lower bound for the new TSP then S Lower bound for new TSP = = + = 9 If the two vertices connecting Syd's home to the tree were and then there would be a Hamiltonian cycle of length this could be a solution to this TSP and so the best lower bound that would be possible. MT xam 0 Solutions Page of

7 MT xamination 0 Part Part Solutions This essay is just one version of what could be produced; it is probably more detailed than would be expected and is included to provide an overview of the topic as well as an example. Q graph is planar if it can be drawn on a flat piece of paper so that none of the edges intersect each other, consequently the edges will only meet at vertices. If this cannot be done then the graph is non planar. are must be taken, since it is quite possible to have a drawing of a graph in which some edges intersect but for which, on rearrangement of the edges while keeping them incident to the same vertices, the intersections can be removed; such a graph is planar. n important result that applies to all planar graphs is uler s formula which states that v e + f =, where v is the number of vertices, e is the number of edges and f is the number of faces. It is possible to deduce from uler s formula two inequalities which apply respectively to simple connected planar graphs with three or more vertices and to simple connected planar graphs with three or more vertices and no triangles. These inequalities put limits on the number of edges such graphs can have and allow us to deduce that the complete graph K and the complete bipartite graph K, are both non planar. The inequalities referred to above do not provide a complete characterisation of a planar graph since some graphs, such as the Petersen graph, satisfy both of the inequalities and yet are non planar. However, there is a criterion by which we can characterise all planar graphs and hence all non planar graphs, this is Kuratowski s theorem. This theorem states that a graph is planar if and only if it does not contain a subdivision of K or K,. subdivision of a graph is a graph that can be obtained from the given graph by the insertion of one or more vertices of degree two on any edges. learly the insertion (or removal) of such vertices will not change the planarity or otherwise of the given graph. Thus in a sense K and K, are basic non planar components for any non planar graph and a planar graph cannot contain either of them as a subgraph or as a subdivision. Kuratowski s theorem, although characterising planar graphs completely, does not provide an easy means of testing whether a given graph is planar. There are efficient algorithms for doing this but one simple test which can be applied to any graph, which is not too large and which contains a Hamiltonian cycle, is the cycle method. This algorithm proceeds by specifying a Hamiltonian cycle in the graph and then attempts to allocate all the edges of the graph, not in the cycle, to two sets of non intersecting edges. If this can be done, then one set can be drawn inside the cycle and the other outside the cycle and the graph is planar. If there is a conflict and the edges do not fall into two disjoint sets then the graph is non planar. Given any planar graph, it is possible to construct another graph called its dual by placing a vertex in each face and joining these vertices by edges so that one edge of the dual crosses each edge of the original. The dual of a planar graph is planar but is not necessarily unique; however, the dual of a dual is a graph isomorphic to the original graph. There is a one to one correspondence between edges in a graph and its dual, vertices of one correspond to faces of the other and cycles in one correspond to cutsets in the other, so uler s theorem still applies. Knowing whether a graph is planar or knowing how to decompose a given graph into two or more planar subgraphs is important in various applications. One such application is the printed circuit board problem in which electronic components must be connected by conducting strips which must not cross on a particular board. MT xam 0 Solutions Page of

8 MT xamination 0 Part Part Solutions Q. S H T J G (a) S G H J 0 * T * * (I) The next labels to be assigned are *, G, H, J (ii) The next potential to be assigned is G (iii) The remaining vertex potentials are, J 9, H 0, T. (iv) The shortest paths are of length, SHT (red), SGJT (green), SGT (blue) and SHT S G H J T S S S, S,,,,,,,, G, G, H, J 0,0 0,9 9,,,,, (i) The potentials assigned at the next step are 0, and G 9, (ii) The remaing potentials are H, J, T. (iii) The single longest path of length is SJT (c) If one arc is to be increased by (i) If the arc is S all the shortest paths are increased by to length, but there are more shortest paths since now S = S (SHT, SGJT, SGT) the longest path is unchanged (ii) If the arc is J all the shortest paths are unchanged the longest path is increased by to length 9 (iii) If the arc is JT the shortest path is still of length but SGJT is no longer one of them the longest path is increased by to length 9 MT xam 0 Solutions Page of

9 MT xamination 0 Q xtra Part Part Solutions or those who wish to see the full tables for both algorithms (a) The shortest path algorithm S G H J T 0 * * * The longest path algorithm S G H J T S S S, S,,,,,,,, G, G, H, J 0,,,,,,0,9 0 9,,,,, MT xam 0 Solutions Page 9 of

10 MT xamination 0 Part Part Solutions Q (a) (i) (ii) The set {,,, } can only cover the tutorials {,, } Since there is a set of tutors larger than the set of tutorials covered by those tutors, The Marriage Theorem states that a complete matching of tutors to tutorials is impossible. (iii) The maximum number of tutors that can be assigned to tutorials is. The allocation,,,, is such an allocation. (i) () () () () () () () () (G) breakthrough () G (G) () H (H) lternating path HG giving an improved matching as follows () () () () breakthrough () () () () () () () G () H (G) lternating path Maximum matching,,,,,, G, H (ii) No, the maximum matching is not unique there are the other has and G all others the same. MT xam 0 Solutions Page of

11 MT xamination 0 Part Part Solutions This essay is just one version of what could be produced; it is more detailed than would be expected and is included to provide an overview of the topic as well as an example. Q. It might seem that an electrical network should be modelled by a mathematical network, but this is not the case. When setting up the model at the outset, before any analysis has taken place, we do not have a predetermined direction associated with the current flow through, or the voltage across, any particular component. Thus the directions must be left to be determined later and a graph is the appropriate model, albeit with an arbitrary direction associated with each edge, so producing an oriented graph The two main aspects of an electrical network which require modelling are the components and their interconnections. The components can be of various kinds and are connected to other components through their terminals which can number two or more. Two terminal components are represented by a single edge, three terminal components by two edges and in general n terminal components by n edges as explained below. There are two laws used in the graphical representation of any physical network, they are the vertex law and the cycle law. The vertex law states that the algebraic sum of the through variables at any vertex is zero, while the cycle law states that the algebraic sum of the across variables around any cycle is zero. In the context of electrical networks the through variable is current, the across variable is voltage and the laws are generally known as Kirchoff s Laws. These laws allow the connectivity or topology of the network to be described in terms of a set of fundamental equations called the fundamental cycle and cutset equations. These equations are not unique for any network but are defined in terms of a spanning tree for the oriented graph representing the network. That these equations are only dependent on the interconnections of the components can be seen from the fact that they can be obtained directly from the incidence matrix for the oriented graph, which alone fully describes the connectivity. s mentioned above, a three terminal component is represented by two edges. It might seem obvious that we should have three edges connecting each terminal to every other so having six variables (three through and three across) to represent the component. Kirchoff s laws, however, generate two equations which connect these six variables so reducing the number of independent variables by two and hence the number of edges required by one. In a similar way every multi-terminal component with n terminals has the number of edges in its graphical representation reduced by the equations generated by Kirchoff s laws so producing in every case a representation by a tree with n edges. These representations are by no means unique. The representation of electrical networks by graphs allows large and complex systems to be analysed through the simple ideas of graph theory and this facilitates the use of computers to solve large and complex electrical network problems. There are also other benefits from the use of graph theory. contrasting and alternative viewpoint allows ideas and concepts from graph theory to illuminate those in electrical network theory. simple example is the idea of the dual of a planar graph as defined in graph theory. When transferred to the study of electrical networks this introduces the idea of the dual of a planar electrical network. The benefit is that if we have solved an electrical network problem, we can immediately write down the solution of the corresponding dual network problem by just using the correspondences indicated by duality. MT xam 0 Solutions Page of

12 MT xamination 0 Part Part Solutions Q (a) (i) The system S has binary joints, ternary joints and binary links (ii) The direct graph representation of S is shown above next to the system (iii) M = g n = x = = This is the mobility criterion for graphs which applies to all systems with only binary links but with joints of any multiplicity. This expansion uses the links indicated by circled crosses (i) The interchange graph of the above system S is shown above next to the system (ii) This expansion uses the links indicated by circled crosses different expansion S of the system is given above with its interchange graph next to the system (iii) (c) These two interchange graphs are not isomorphic because the second contains a vertex of degree and no vertices of degree while the first contains two vertices of degree and none of degree There are four distinct non-isomorphic interchange graphs resulting from the expansions of the ternary joints of S. part from those above there are two more as follows Not isomorphic to that in This expansion because the two vertices uses the links of degree are joined by an indicated by edge while in the two circled crosses vertices of degree are not joined. Not isomorphic to that in This expansion because the two vertices uses the links of degree are joined to only indicated by one common vertex of degree circled crosses while in the two vertices of degree have two vertices of degree in common ue to the symmetry of the given system there are no other choices for the links to be used to perform the expansion possible see the diagrams with the circled crosses. MT xam 0 Solutions Page of

13 MT xamination 0 Part Part Solutions Q9 G H I J K L M (a) (i) v =, b =, r =, k = (ii) r(k ) ( ) λ= = = = v (iii) or a finite projective plane v = b = n + n + =, r = k = n + = so n = (i) No, Δ is not resolvable since for a replicate containing the block (0,,, 9) we cannot find a block containing since (0,,, ) contains 0 and (,,, ) contains while H (,,, ) contains and finally I(,,, 9) contains 9 (ii) Yes, Δ is a cyclic design as indicated below by rearranging the blocks H J I K L M G (c) If is the complement of Δ then (i) v= v=, b= b=, k= v k= = 9, r= b r= = 9 (ii) y Th. λ= b r+λ= + = the design is balanced (d) (i) r = the number of cards carrying a particular symbol k = the number of symbols on a particular card (ii) finite projective plane of order n = has + + = points and the same number of lines. There would be r = k = n + = different symbols and each would occur on cards in the full set giving x = symbols Since in a finite projective plane any two lines (cards) have only one common point (symbol) there will be one symbol which occurs on only cards and symbols which occur on cards while the remaining symbols occur on eight cards ( x + x + x = = 0 = x = - ) MT xam 0 Solutions Page of

14 MT xamination 0 Part Part Solutions This essay is just one version of what could be produced; it is probably more detailed than would be expected and is included to provide an overview of the topic as well as an example. Q0 linear binary code is one in which the sum x + y of any two codewords x and y is also a codeword. The operations are performed modulo since the codewords are strings of s and 0 s, elements from the finite field Z = ({0, }, +, ). If the codewords are of length n such as x = x x x n, where x i {0, } and there are k of them, then the code is called an (n, k) linear code. The k codewords form a subset of all possible binary words of length n, which themselves form an n- dimensional vector space consisting of all n n-tuples. Thus a linear code is a k-dimensional subspace of an n-dimensional vector space. very linear code contains the zero codeword 0 consisting of n zeros. Since codewords of an (n, k) linear code form a k-dimensional sub-space, it is possible to find a set of k of them which form a basis for this sub-space. ll codewords can be written as linear combinations of the basic set. Such a set of k codewords is called a generator set for the code and can be written as a k n matrix G, the generator matrix for the code. very codeword is formed by a linear combination of the rows of G, there being k such combinations. ll codewords in a linear code have a weight defined as the number of s in the codeword. The smallest weight of any non-zero codeword is the minimum distance δ for the code and this indicates the error detecting and correcting properties of the code. So we use (n, k, δ) to specify the code. Related to this is a matrix H for which Hx = 0 for all codewords x. This matrix is called the parity check matrix because the n k rows of H specify combinations of the bits of the codewords which must always be zero for all codewords x. Parity check matrices are used in decoding received binary words. Multiplication of a received word by the parity check matrix produces a vector called an error syndrome which, if the code is a single error correcting code and the received word contains just one error, will be a column of the parity check matrix. The position of this column will indicate the position of the error in the received word. In a similar way, the error syndrome will be a linear combination of m columns from the parity check matrix if the code is an m error correcting code and the received word contains m errors. Two linear codes are equivalent if the bits in all the codewords of one can be rearranged by some fixed rule to produce all the codewords of the other. If this is the case, then the columns of the generator matrix of one code must be related by rearrangement to the columns of the generator matrix of the other by the same fixed rule. Hence every (n, k) linear code is equivalent to a code for which the generator matrix is in standard form with a unit sub matrix in the first k columns. If we take the rows of the parity check matrix of a code as the generator set of a new code we obtain the code words of a code called the dual code * of the code, so G* = H. lso the generator matrix of is the parity check matrix of * so H* = G. or any linear code we can produce another code called the xtended ode by adding an extra digit to make all codewords even parity. The Hamming codes are a set of single error correcting codes formed so the parity check matrix is a set of columns which are the m non-zero binary numbers of length m. Hamming odes are ( m, m m, ) codes. If we have a t error correcting code for which every binary word is within a distance t of a codeword the code is called perfect because there are no binary words that cannot be decoded unambiguously. The Hamming codes are all perfect codes. cyclic code is one in which, whenever x = x x x x n is a codeword then so is x = x x x n x. There may be several disjoint cycles in a cyclic code. Such codes are important since they can be practically implemented in a simple way. In conclusion many linear codes are of practical importance. The Reed-Muller codes have been used for transmission from the Mariner 9 space probe because of their ability to correct many errors, while the Reed-Solomon codes are used for digital music on s because of their ability to correct bursts of errors as produced by scratches. MT xam 0 Solutions Page of

15 MT xamination 0 Part Part Solutions ssay Outlines Q. Introduction the definition of a planar graph. uler s formula for planar graphs resulting inequalities. Kuratowski s Theorem subdivisions of a graph. Testing for planarity Hamiltonian cycle method. uality of planar graphs cycles and cutsets. onclusion importance of planar graphs for printed circuit boards. Q. Introduction - the use of oriented graphs to model networks The vertex and cycle laws for through and across variables respectively undamental cycles and cutsets Incidence matrices The representation of n-terminal components by trees onclusion dual graphs and dual networks Q0. Introduction definition of a linear binary code Generator set of codewords Generator matrices rror orrection and etection Parity heck matrices quivalent odes, ual odes and xtended odes Hamming odes, Perfect odes and yclic codes onclusion Reed - Muller and Reed - Solomon odes MT xam 0 Solutions Page of

Part 1. Twice the number of edges = 2 9 = 18. Thus the Handshaking lemma... The sum of the vertex degrees = twice the number of edges holds.

Part 1. Twice the number of edges = 2 9 = 18. Thus the Handshaking lemma... The sum of the vertex degrees = twice the number of edges holds. MT6 Examination 16 Q1 (a) Part 1 Part1 Solutions (b) egree Sequence (,,,,, ) (c) Sum of the vertex degrees = + + + + + = 18 Twice the number of edges = 9 = 18. Thus the Handshaking lemma... The sum of

More information

MT365 Examination 2007 Part 1. Q1 (a) (b) (c) A

MT365 Examination 2007 Part 1. Q1 (a) (b) (c) A MT6 Examination Part Solutions Q (a) (b) (c) F F F E E E G G G G is both Eulerian and Hamiltonian EF is both an Eulerian trail and a Hamiltonian cycle. G is Hamiltonian but not Eulerian and EF is a Hamiltonian

More information

Graph and Digraph Glossary

Graph and Digraph Glossary 1 of 15 31.1.2004 14:45 Graph and Digraph Glossary A B C D E F G H I-J K L M N O P-Q R S T U V W-Z Acyclic Graph A graph is acyclic if it contains no cycles. Adjacency Matrix A 0-1 square matrix whose

More information

Part II. Graph Theory. Year

Part II. Graph Theory. Year Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 53 Paper 3, Section II 15H Define the Ramsey numbers R(s, t) for integers s, t 2. Show that R(s, t) exists for all s,

More information

Elements of Graph Theory

Elements of Graph Theory Elements of Graph Theory Quick review of Chapters 9.1 9.5, 9.7 (studied in Mt1348/2008) = all basic concepts must be known New topics we will mostly skip shortest paths (Chapter 9.6), as that was covered

More information

Further Mathematics 2016 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks

Further Mathematics 2016 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Further Mathematics 2016 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Key knowledge the conventions, terminology, properties and types of graphs; edge, face, loop,

More information

GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS

GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS GRAPHS, GRAPH MODELS, GRAPH TERMINOLOGY, AND SPECIAL TYPES OF GRAPHS DR. ANDREW SCHWARTZ, PH.D. 10.1 Graphs and Graph Models (1) A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes)

More information

CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS

CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1. Define Graph. A graph G = (V, E) consists

More information

Discrete mathematics II. - Graphs

Discrete mathematics II. - Graphs Emil Vatai April 25, 2018 Basic definitions Definition of an undirected graph Definition (Undirected graph) An undirected graph or (just) a graph is a triplet G = (ϕ, E, V ), where V is the set of vertices,

More information

4. (a) Draw the Petersen graph. (b) Use Kuratowski s teorem to prove that the Petersen graph is non-planar.

4. (a) Draw the Petersen graph. (b) Use Kuratowski s teorem to prove that the Petersen graph is non-planar. UPPSALA UNIVERSITET Matematiska institutionen Anders Johansson Graph Theory Frist, KandMa, IT 010 10 1 Problem sheet 4 Exam questions Solve a subset of, say, four questions to the problem session on friday.

More information

1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G))

1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G)) 10 Graphs 10.1 Graphs and Graph Models 1. a graph G = (V (G), E(G)) consists of a set V (G) of vertices, and a set E(G) of edges (edges are pairs of elements of V (G)) 2. an edge is present, say e = {u,

More information

MAS 341: GRAPH THEORY 2016 EXAM SOLUTIONS

MAS 341: GRAPH THEORY 2016 EXAM SOLUTIONS MS 41: PH THEOY 2016 EXM SOLUTIONS 1. Question 1 1.1. Explain why any alkane C n H 2n+2 is a tree. How many isomers does C 6 H 14 have? Draw the structure of the carbon atoms in each isomer. marks; marks

More information

Module 2: NETWORKS AND DECISION MATHEMATICS

Module 2: NETWORKS AND DECISION MATHEMATICS Further Mathematics 2017 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Key knowledge the conventions, terminology, properties and types of graphs; edge, face, loop,

More information

Planar Graph (7A) Young Won Lim 5/21/18

Planar Graph (7A) Young Won Lim 5/21/18 Planar Graph (7A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later

More information

5 Graphs

5 Graphs 5 Graphs jacques@ucsd.edu Some of the putnam problems are to do with graphs. They do not assume more than a basic familiarity with the definitions and terminology of graph theory. 5.1 Basic definitions

More information

UNIVERSITY OF MANITOBA FINAL EXAMINATION TITLE PAGE TIME: 3 hours EXAMINER: M. Davidson. DATE: April 14, 2012

UNIVERSITY OF MANITOBA FINAL EXAMINATION TITLE PAGE TIME: 3 hours EXAMINER: M. Davidson. DATE: April 14, 2012 TITL PG MILY NM: (Print in ink) GIVN NM(S): (Print in ink) STUDNT NUMBR: ST NUMBR: SIGNTUR: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDNTS: This is a hour exam. Please

More information

Planar Graph (7A) Young Won Lim 6/20/18

Planar Graph (7A) Young Won Lim 6/20/18 Planar Graph (7A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. An Introduction to Graph Theory

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. An Introduction to Graph Theory SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics An Introduction to Graph Theory. Introduction. Definitions.. Vertices and Edges... The Handshaking Lemma.. Connected Graphs... Cut-Points and Bridges.

More information

MATH 350 GRAPH THEORY & COMBINATORICS. Contents

MATH 350 GRAPH THEORY & COMBINATORICS. Contents MATH 350 GRAPH THEORY & COMBINATORICS PROF. SERGEY NORIN, FALL 2013 Contents 1. Basic definitions 1 2. Connectivity 2 3. Trees 3 4. Spanning Trees 3 5. Shortest paths 4 6. Eulerian & Hamiltonian cycles

More information

Assignment 4 Solutions of graph problems

Assignment 4 Solutions of graph problems Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of

More information

WUCT121. Discrete Mathematics. Graphs

WUCT121. Discrete Mathematics. Graphs WUCT121 Discrete Mathematics Graphs WUCT121 Graphs 1 Section 1. Graphs 1.1. Introduction Graphs are used in many fields that require analysis of routes between locations. These areas include communications,

More information

Introduction III. Graphs. Motivations I. Introduction IV

Introduction III. Graphs. Motivations I. Introduction IV Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg

More information

Assignment 1 Introduction to Graph Theory CO342

Assignment 1 Introduction to Graph Theory CO342 Assignment 1 Introduction to Graph Theory CO342 This assignment will be marked out of a total of thirty points, and is due on Thursday 18th May at 10am in class. Throughout the assignment, the graphs are

More information

Topic 10 Part 2 [474 marks]

Topic 10 Part 2 [474 marks] Topic Part 2 [474 marks] The complete graph H has the following cost adjacency matrix Consider the travelling salesman problem for H a By first finding a minimum spanning tree on the subgraph of H formed

More information

MAS341 Graph Theory 2015 exam solutions

MAS341 Graph Theory 2015 exam solutions MAS4 Graph Theory 0 exam solutions Question (i)(a) Draw a graph with a vertex for each row and column of the framework; connect a row vertex to a column vertex if there is a brace where the row and column

More information

Network Topology and Graph

Network Topology and Graph Network Topology Network Topology and Graph EEE442 Computer Method in Power System Analysis Any lumped network obeys 3 basic laws KVL KCL linear algebraic constraints Ohm s law Anawach Sangswang Dept.

More information

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI Department of Computer Science and Engineering CS6702 - GRAPH THEORY AND APPLICATIONS Anna University 2 & 16 Mark Questions & Answers Year / Semester: IV /

More information

Fundamental Properties of Graphs

Fundamental Properties of Graphs Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,

More information

Basics of Graph Theory

Basics of Graph Theory Basics of Graph Theory 1 Basic notions A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Simple graphs have their

More information

Matching and Planarity

Matching and Planarity Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of

More information

Combinatorics Summary Sheet for Exam 1 Material 2019

Combinatorics Summary Sheet for Exam 1 Material 2019 Combinatorics Summary Sheet for Exam 1 Material 2019 1 Graphs Graph An ordered three-tuple (V, E, F ) where V is a set representing the vertices, E is a set representing the edges, and F is a function

More information

GRAPHS: THEORY AND ALGORITHMS

GRAPHS: THEORY AND ALGORITHMS GRAPHS: THEORY AND ALGORITHMS K. THULASIRAMAN M. N. S. SWAMY Concordia University Montreal, Canada A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Brisbane / Toronto /

More information

Graph theory. Po-Shen Loh. June We begin by collecting some basic facts which can be proved via bare-hands techniques.

Graph theory. Po-Shen Loh. June We begin by collecting some basic facts which can be proved via bare-hands techniques. Graph theory Po-Shen Loh June 013 1 Basic results We begin by collecting some basic facts which can be proved via bare-hands techniques. 1. The sum of all of the degrees is equal to twice the number of

More information

CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK

CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK 1. Define Graph. 2. Define Simple graph. 3. Write few problems

More information

Mathematics and Statistics, Part A: Graph Theory Problem Sheet 1, lectures 1-4

Mathematics and Statistics, Part A: Graph Theory Problem Sheet 1, lectures 1-4 1. Draw Mathematics and Statistics, Part A: Graph Theory Problem Sheet 1, lectures 1-4 (i) a simple graph. A simple graph has a non-empty vertex set and no duplicated edges. For example sketch G with V

More information

Computing Linkless and Flat Embeddings of Graphs in R 3

Computing Linkless and Flat Embeddings of Graphs in R 3 Computing Linkless and Flat Embeddings of Graphs in R 3 Stephan Kreutzer Technical University Berlin based on joint work with Ken-ichi Kawarabayashi, Bojan Mohar and Bruce Reed Graph Theory @ Georgie Tech

More information

Basic Combinatorics. Math 40210, Section 01 Fall Homework 4 Solutions

Basic Combinatorics. Math 40210, Section 01 Fall Homework 4 Solutions Basic Combinatorics Math 40210, Section 01 Fall 2012 Homework 4 Solutions 1.4.2 2: One possible implementation: Start with abcgfjiea From edge cd build, using previously unmarked edges: cdhlponminjkghc

More information

Chapter 2 Graphs. 2.1 Definition of Graphs

Chapter 2 Graphs. 2.1 Definition of Graphs Chapter 2 Graphs Abstract Graphs are discrete structures that consist of vertices and edges connecting some of these vertices. Graphs have many applications in Mathematics, Computer Science, Engineering,

More information

0.0.1 Network Analysis

0.0.1 Network Analysis Graph Theory 0.0.1 Network Analysis Prototype Example: In Algonquian Park the rangers have set up snowmobile trails with various stops along the way. The system of trails is our Network. The main entrance

More information

Math 443/543 Graph Theory Notes 5: Planar graphs and coloring

Math 443/543 Graph Theory Notes 5: Planar graphs and coloring Math 443/543 Graph Theory Notes 5: Planar graphs and coloring David Glickenstein October 10, 2014 1 Planar graphs The Three Houses and Three Utilities Problem: Given three houses and three utilities, can

More information

Characterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)

Characterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4) S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G

More information

Definition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1

Definition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1 Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,

More information

Planar Graphs. 1 Graphs and maps. 1.1 Planarity and duality

Planar Graphs. 1 Graphs and maps. 1.1 Planarity and duality Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter

More information

An Investigation of the Planarity Condition of Grötzsch s Theorem

An Investigation of the Planarity Condition of Grötzsch s Theorem Le Chen An Investigation of the Planarity Condition of Grötzsch s Theorem The University of Chicago: VIGRE REU 2007 July 16, 2007 Abstract The idea for this paper originated from Professor László Babai

More information

Introduction to. Graph Theory. Second Edition. Douglas B. West. University of Illinois Urbana. ftentice iiilil PRENTICE HALL

Introduction to. Graph Theory. Second Edition. Douglas B. West. University of Illinois Urbana. ftentice iiilil PRENTICE HALL Introduction to Graph Theory Second Edition Douglas B. West University of Illinois Urbana ftentice iiilil PRENTICE HALL Upper Saddle River, NJ 07458 Contents Preface xi Chapter 1 Fundamental Concepts 1

More information

Definition 1.1. A matching M in a graph G is called maximal if there is no matching M in G so that M M.

Definition 1.1. A matching M in a graph G is called maximal if there is no matching M in G so that M M. 1 Matchings Before, we defined a matching as a set of edges no two of which share an end in common. Suppose that we have a set of jobs and people and we want to match as many jobs to people as we can.

More information

Assignments are handed in on Tuesdays in even weeks. Deadlines are:

Assignments are handed in on Tuesdays in even weeks. Deadlines are: Tutorials at 2 3, 3 4 and 4 5 in M413b, on Tuesdays, in odd weeks. i.e. on the following dates. Tuesday the 28th January, 11th February, 25th February, 11th March, 25th March, 6th May. Assignments are

More information

Graph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow

Graph Theory. Connectivity, Coloring, Matching. Arjun Suresh 1. 1 GATE Overflow Graph Theory Connectivity, Coloring, Matching Arjun Suresh 1 1 GATE Overflow GO Classroom, August 2018 Thanks to Subarna/Sukanya Das for wonderful figures Arjun, Suresh (GO) Graph Theory GATE 2019 1 /

More information

Hamiltonian cycles in bipartite quadrangulations on the torus

Hamiltonian cycles in bipartite quadrangulations on the torus Hamiltonian cycles in bipartite quadrangulations on the torus Atsuhiro Nakamoto and Kenta Ozeki Abstract In this paper, we shall prove that every bipartite quadrangulation G on the torus admits a simple

More information

PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS

PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PAUL BALISTER Abstract It has been shown [Balister, 2001] that if n is odd and m 1,, m t are integers with m i 3 and t i=1 m i = E(K n) then K n can be decomposed

More information

Introduction to Graph Theory

Introduction to Graph Theory Introduction to Graph Theory George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 351 George Voutsadakis (LSSU) Introduction to Graph Theory August 2018 1 /

More information

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95

Algorithms: Graphs. Amotz Bar-Noy. Spring 2012 CUNY. Amotz Bar-Noy (CUNY) Graphs Spring / 95 Algorithms: Graphs Amotz Bar-Noy CUNY Spring 2012 Amotz Bar-Noy (CUNY) Graphs Spring 2012 1 / 95 Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Amotz

More information

6. Lecture notes on matroid intersection

6. Lecture notes on matroid intersection Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans May 2, 2017 6. Lecture notes on matroid intersection One nice feature about matroids is that a simple greedy algorithm

More information

CMSC Honors Discrete Mathematics

CMSC Honors Discrete Mathematics CMSC 27130 Honors Discrete Mathematics Lectures by Alexander Razborov Notes by Justin Lubin The University of Chicago, Autumn 2017 1 Contents I Number Theory 4 1 The Euclidean Algorithm 4 2 Mathematical

More information

Algorithms. Graphs. Algorithms

Algorithms. Graphs. Algorithms Algorithms Graphs Algorithms Graphs Definition: A graph is a collection of edges and vertices. Each edge connects two vertices. Algorithms 1 Graphs Vertices: Nodes, points, computers, users, items,...

More information

Introductory Combinatorics

Introductory Combinatorics Introductory Combinatorics Third Edition KENNETH P. BOGART Dartmouth College,. " A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Toronto Sydney Tokyo xm CONTENTS

More information

Graph Theory. Part of Texas Counties.

Graph Theory. Part of Texas Counties. Graph Theory Part of Texas Counties. We would like to visit each of the above counties, crossing each county only once, starting from Harris county. Is this possible? This problem can be modeled as a graph.

More information

Chapter 1 Graph Theory

Chapter 1 Graph Theory Chapter Graph Theory - Representations of Graphs Graph, G=(V,E): It consists of the set V of vertices and the set E of edges. If each edge has its direction, the graph is called the directed graph (digraph).

More information

An Introduction to Graph Theory

An Introduction to Graph Theory An Introduction to Graph Theory CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 12:00, Friday 17 th February 2012 Outline 1 Graphs 2 Paths and cycles 3 Graphs and

More information

Lecture 5: Graphs. Rajat Mittal. IIT Kanpur

Lecture 5: Graphs. Rajat Mittal. IIT Kanpur Lecture : Graphs Rajat Mittal IIT Kanpur Combinatorial graphs provide a natural way to model connections between different objects. They are very useful in depicting communication networks, social networks

More information

Chapter 3: Paths and Cycles

Chapter 3: Paths and Cycles Chapter 3: Paths and Cycles 5 Connectivity 1. Definitions: Walk: finite sequence of edges in which any two consecutive edges are adjacent or identical. (Initial vertex, Final vertex, length) Trail: walk

More information

Lecture 3: Recap. Administrivia. Graph theory: Historical Motivation. COMP9020 Lecture 4 Session 2, 2017 Graphs and Trees

Lecture 3: Recap. Administrivia. Graph theory: Historical Motivation. COMP9020 Lecture 4 Session 2, 2017 Graphs and Trees Administrivia Lecture 3: Recap Assignment 1 due 23:59 tomorrow. Quiz 4 up tonight, due 15:00 Thursday 31 August. Equivalence relations: (S), (R), (T) Total orders: (AS), (R), (T), (L) Partial orders: (AS),

More information

Lecture 4: Bipartite graphs and planarity

Lecture 4: Bipartite graphs and planarity Lecture 4: Bipartite graphs and planarity Anders Johansson 2011-10-22 lör Outline Bipartite graphs A graph G is bipartite with bipartition V1, V2 if V = V1 V2 and all edges ij E has one end in V1 and V2.

More information

Graph Theory Mini-course

Graph Theory Mini-course Graph Theory Mini-course Anthony Varilly PROMYS, Boston University, Boston, MA 02215 Abstract Intuitively speaking, a graph is a collection of dots and lines joining some of these dots. Many problems in

More information

Math Summer 2012

Math Summer 2012 Math 481 - Summer 2012 Final Exam You have one hour and fifty minutes to complete this exam. You are not allowed to use any electronic device. Be sure to give reasonable justification to all your answers.

More information

SAMPLE. MODULE 5 Undirected graphs

SAMPLE. MODULE 5 Undirected graphs H P T R MOUL Undirected graphs How do we represent a graph by a diagram and by a matrix representation? How do we define each of the following: graph subgraph vertex edge (node) loop isolated vertex bipartite

More information

Lecture 20 : Trees DRAFT

Lecture 20 : Trees DRAFT CS/Math 240: Introduction to Discrete Mathematics 4/12/2011 Lecture 20 : Trees Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we discussed graphs. Today we continue this discussion,

More information

Discrete mathematics

Discrete mathematics Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 470-2301/02, Winter term 2017/2018 About this file This file is meant to be a guideline for the lecturer. Many

More information

Number Theory and Graph Theory

Number Theory and Graph Theory 1 Number Theory and Graph Theory Chapter 7 Graph properties By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 2 Module-2: Eulerian

More information

CS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs

CS 311 Discrete Math for Computer Science Dr. William C. Bulko. Graphs CS 311 Discrete Math for Computer Science Dr. William C. Bulko Graphs 2014 Definitions Definition: A graph G = (V,E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge

More information

Lecture 6: Graph Properties

Lecture 6: Graph Properties Lecture 6: Graph Properties Rajat Mittal IIT Kanpur In this section, we will look at some of the combinatorial properties of graphs. Initially we will discuss independent sets. The bulk of the content

More information

Exercise set 2 Solutions

Exercise set 2 Solutions Exercise set 2 Solutions Let H and H be the two components of T e and let F E(T ) consist of the edges of T with one endpoint in V (H), the other in V (H ) Since T is connected, F Furthermore, since T

More information

Varying Applications (examples)

Varying Applications (examples) Graph Theory Varying Applications (examples) Computer networks Distinguish between two chemical compounds with the same molecular formula but different structures Solve shortest path problems between cities

More information

Lecture 1: Examples, connectedness, paths and cycles

Lecture 1: Examples, connectedness, paths and cycles Lecture 1: Examples, connectedness, paths and cycles Anders Johansson 2011-10-22 lör Outline The course plan Examples and applications of graphs Relations The definition of graphs as relations Connectedness,

More information

Error-Correcting Codes

Error-Correcting Codes Error-Correcting Codes Michael Mo 10770518 6 February 2016 Abstract An introduction to error-correcting codes will be given by discussing a class of error-correcting codes, called linear block codes. The

More information

Symmetric Product Graphs

Symmetric Product Graphs Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 5-20-2015 Symmetric Product Graphs Evan Witz Follow this and additional works at: http://scholarworks.rit.edu/theses

More information

v 1 v 2 r 3 r 4 v 3 v 4 Figure A plane embedding of K 4.

v 1 v 2 r 3 r 4 v 3 v 4 Figure A plane embedding of K 4. Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities

More information

GRAPH DECOMPOSITION BASED ON DEGREE CONSTRAINTS. March 3, 2016

GRAPH DECOMPOSITION BASED ON DEGREE CONSTRAINTS. March 3, 2016 GRAPH DECOMPOSITION BASED ON DEGREE CONSTRAINTS ZOÉ HAMEL March 3, 2016 1. Introduction Let G = (V (G), E(G)) be a graph G (loops and multiple edges not allowed) on the set of vertices V (G) and the set

More information

Three applications of Euler s formula. Chapter 10

Three applications of Euler s formula. Chapter 10 Three applications of Euler s formula Chapter 10 A graph is planar if it can be drawn in the plane R without crossing edges (or, equivalently, on the -dimensional sphere S ). We talk of a plane graph if

More information

Planarity: dual graphs

Planarity: dual graphs : dual graphs Math 104, Graph Theory March 28, 2013 : dual graphs Duality Definition Given a plane graph G, the dual graph G is the plane graph whose vtcs are the faces of G. The correspondence between

More information

Course Introduction / Review of Fundamentals of Graph Theory

Course Introduction / Review of Fundamentals of Graph Theory Course Introduction / Review of Fundamentals of Graph Theory Hiroki Sayama sayama@binghamton.edu Rise of Network Science (From Barabasi 2010) 2 Network models Many discrete parts involved Classic mean-field

More information

About the Tutorial. Audience. Prerequisites. Disclaimer & Copyright. Graph Theory

About the Tutorial. Audience. Prerequisites. Disclaimer & Copyright. Graph Theory About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability,

More information

List of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2,

List of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2, List of Theorems Mat 416, Introduction to Graph Theory 1. Ramsey s Theorem for graphs 8.3.11. Theorem 1 The numbers R(p, q) exist and for p, q 2, R(p, q) R(p 1, q) + R(p, q 1). If both summands on the

More information

UNDIRECTED GRAPH: a set of vertices and a set of undirected edges each of which is associated with a set of one or two of these vertices.

UNDIRECTED GRAPH: a set of vertices and a set of undirected edges each of which is associated with a set of one or two of these vertices. Graphs 1 Graph: A graph G = (V, E) consists of a nonempty set of vertices (or nodes) V and a set of edges E. Each edge has either one or two vertices associated with it, called its endpoints. An edge is

More information

Parallel and perspective projections such as used in representing 3d images.

Parallel and perspective projections such as used in representing 3d images. Chapter 5 Rotations and projections In this chapter we discuss Rotations Parallel and perspective projections such as used in representing 3d images. Using coordinates and matrices, parallel projections

More information

Shannon capacity and related problems in Information Theory and Ramsey Theory

Shannon capacity and related problems in Information Theory and Ramsey Theory Shannon capacity and related problems in Information Theory and Ramsey Theory Eyal Lubetzky Based on Joint work with Noga Alon and Uri Stav May 2007 1 Outline of talk Shannon Capacity of of a graph: graph:

More information

Mock Exam. Juanjo Rué Discrete Mathematics II, Winter Deadline: 14th January 2014 (Tuesday) by 10:00, at the end of the lecture.

Mock Exam. Juanjo Rué Discrete Mathematics II, Winter Deadline: 14th January 2014 (Tuesday) by 10:00, at the end of the lecture. Mock Exam Juanjo Rué Discrete Mathematics II, Winter 2013-2014 Deadline: 14th January 2014 (Tuesday) by 10:00, at the end of the lecture. Problem 1 (2 points): 1. State the definition of perfect graph

More information

Simple graph Complete graph K 7. Non- connected graph

Simple graph Complete graph K 7. Non- connected graph A graph G consists of a pair (V; E), where V is the set of vertices and E the set of edges. We write V (G) for the vertices of G and E(G) for the edges of G. If no two edges have the same endpoints we

More information

Key Graph Theory Theorems

Key Graph Theory Theorems Key Graph Theory Theorems Rajesh Kumar MATH 239 Intro to Combinatorics August 19, 2008 3.3 Binary Trees 3.3.1 Problem (p.82) Determine the number, t n, of binary trees with n edges. The number of binary

More information

1. The following graph is not Eulerian. Make it into an Eulerian graph by adding as few edges as possible.

1. The following graph is not Eulerian. Make it into an Eulerian graph by adding as few edges as possible. 1. The following graph is not Eulerian. Make it into an Eulerian graph by adding as few edges as possible. A graph is Eulerian if it has an Eulerian circuit, which occurs if the graph is connected and

More information

11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions

11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions Introduction Chapter 9 Graph Algorithms graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 2 Definitions an undirected graph G = (V, E) is

More information

Chapter 9 Graph Algorithms

Chapter 9 Graph Algorithms Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 3 Definitions an undirected graph G = (V, E)

More information

Graph Theory Questions from Past Papers

Graph Theory Questions from Past Papers Graph Theory Questions from Past Papers Bilkent University, Laurence Barker, 19 October 2017 Do not forget to justify your answers in terms which could be understood by people who know the background theory

More information

r=1 The Binomial Theorem. 4 MA095/98G Revision

r=1 The Binomial Theorem. 4 MA095/98G Revision Revision Read through the whole course once Make summary sheets of important definitions and results, you can use the following pages as a start and fill in more yourself Do all assignments again Do the

More information

Degree of nonsimple graphs. Chemistry questions. Degree Sequences. Pigeon party.

Degree of nonsimple graphs. Chemistry questions. Degree Sequences. Pigeon party. 1. WEEK 1 PROBLEMS 1.1. Degree of nonsimple graphs. In the lecture notes we defined the degree d(v) of a vertex v to be the number of vertices adjacent to v. To see why Euler s theorem doesn t hold for

More information

Answers to specimen paper questions. Most of the answers below go into rather more detail than is really needed. Please let me know of any mistakes.

Answers to specimen paper questions. Most of the answers below go into rather more detail than is really needed. Please let me know of any mistakes. Answers to specimen paper questions Most of the answers below go into rather more detail than is really needed. Please let me know of any mistakes. Question 1. (a) The degree of a vertex x is the number

More information

The University of Sydney MATH2969/2069. Graph Theory Tutorial 2 (Week 9) 2008

The University of Sydney MATH2969/2069. Graph Theory Tutorial 2 (Week 9) 2008 The University of Sydney MATH99/09 Graph Theory Tutorial (Week 9) 00. Show that the graph on the left is Hamiltonian, but that the other two are not. To show that the graph is Hamiltonian, simply find

More information

Chapter 6 GRAPH COLORING

Chapter 6 GRAPH COLORING Chapter 6 GRAPH COLORING A k-coloring of (the vertex set of) a graph G is a function c : V (G) {1, 2,..., k} such that c (u) 6= c (v) whenever u is adjacent to v. Ifak-coloring of G exists, then G is called

More information

Computer Science 280 Fall 2002 Homework 10 Solutions

Computer Science 280 Fall 2002 Homework 10 Solutions Computer Science 280 Fall 2002 Homework 10 Solutions Part A 1. How many nonisomorphic subgraphs does W 4 have? W 4 is the wheel graph obtained by adding a central vertex and 4 additional "spoke" edges

More information

Math.3336: Discrete Mathematics. Chapter 10 Graph Theory

Math.3336: Discrete Mathematics. Chapter 10 Graph Theory Math.3336: Discrete Mathematics Chapter 10 Graph Theory Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall

More information