# graph theory in discrete mathematics notes

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For which values of \(n\) is the graph planar? There were 24 couples: 6 choices for the girl and 4 choices for the boy. There were 45 couples: \({10 \choose 2}\) since we must choose two of the 10 people to dance together. How many vertices does your new convex polyhedron contain? Explain what graphs can be used to represent these situations. This gives a total of 57, which is exactly twice the number of edges, since each edge borders exactly 2 faces. Its two neighbors (adjacent to the blue pentagon) get colored green. \( \def\circleB{(.5,0) circle (1)}\) \def\Q{\mathbb Q} \def\iffmodels{\bmodels\models} The quiz is based on my lectures notes (pages … \def\F{\mathbb F} Yes. Our Discrete mathematics Structure Tutorial is designed for beginners and professionals both. Functions 27 2.3. \draw (\x,\y) +(90:\r) -- +(30:\r) -- +(-30:\r) -- +(-90:\r) -- +(-150:\r) -- +(150:\r) -- cycle; \(\def\d{\displaystyle} ), The chromatic number of \(K_{3,4}\) is 2, since the graph is bipartite. \( \def\C{\mathbb C}\) Are you? \( \def\d{\displaystyle}\) Most discrete books put logic ï¬rst as a preliminary, which certainly has its advantages. Notes on Discrete Mathematics Miguel A. Lerma. \draw (\x,\y) node{#3}; \( \def\circleClabel{(.5,-2) node[right]{$C$}}\) Graphs are made up of a collection of dots called verticesand lines connecting those dots called edges. These basic concepts of sets, logic functions and graph theory are applied to Boolean Algebra and logic networks while the advanced concepts of functions and algebraic â¦ What is the smallest value of \(n\) for which the graph might be planar? Some graphs occur frequently enough in graph theory that they deserve special mention. Every bipartite graph has chromatic number 2. Discrete Mathematics is the mathematics of computing discrete elements using algebra and arithmetic.The use of discrete mathematics is increasing as it can be easily applied in the fields of mathematics and arithmetic. Is it possible to color the vertices of the graph so that related vertices have different colors using a small number of colors? What the objects are and what “related” means varies on context, and this leads to many applications of graph theory to science and other areas of math. Consider a “different” problem: Below is a drawing of four dots connected by some lines. Is the graph bipartite? Problem 1; Problem 2; Problem 3 & 4; Combinatorics. Every polyhedron can be represented as a planar graph, and the Four Color Theorem says that every planar graph has chromatic number at most 4. Euclid is friends with everyone. \draw (\x,\y) +(90:\r) -- +(30:\r) -- +(-30:\r) -- +(-90:\r) -- +(-150:\r) -- +(150:\r) -- cycle; Is it possible to trace over each line once and only once (without lifting up your pencil, starting and ending on a dot)? Explain. Contents I Notions and Notation ... First Steps in Graph Theory This lecture introduces Graph Theory, the main subject of the course, and includes some basic definitions as well as a number of standard examples. \( \renewcommand{\v}{\vtx{above}{}}\) At the time, there were two islands in the river Pregel, and 7 bridges connecting the islands to each other and to each bank of the river. The graph is planar. All the graphs are planar. if we traverse a graph such â¦ The islands were connected to the banks of the river by seven bridges (as seen below). We call these points vertices (sometimes also called nodes), and the lines, edges. How many couples danced if every girl dances with every boy? 108. The chromatic number of \(K_{3,4}\) is 2, since the graph is bipartite. \( \def\And{\bigwedge}\) Does \(G\) have an Euler path? This is a course note on discrete mathematics as used in Computer Science. \def\y{-\r*#1-sin{30}*\r*#1} \( \def\circleBlabel{(1.5,.6) node[above]{$B$}}\) \( \newcommand{\f}[1]{\mathfrak #1}\) \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} It does not matter how big the islands are, what the bridges are made out of, if the river contains alligators, etc. \( \def\dom{\mbox{dom}}\) ... Latest issue All issues. \( \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}}\) \def\twosetbox{(-2,-1.5) rectangle (2,1.5)} We get that there must be 10 vertices with degree 4 and 8 with degree 3. \( \def\land{\wedge}\) Here is an example graph. Prove your conjecture from part (a) by induction on the number of vertices. (For instance, can you have a tree with 5 vertices and 7 edges?). Is the converse of the statement true or false? Functions 27 2.3. In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". If the graph is planar, then \(n - \frac{5n}{2} + f = 2\) so there would be \(\frac{4+3n}{2}\) faces. each edge has a direction 7. Proofs 13 Chapter 2. The nice thing about looking at graphs instead of pictures of rivers, islands and bridges is that we now have a mathematical object to study. MA8351 DM Notes. W:= f0;1;2;:::g, the set of whole numbers 4. False. False. Pictures like the dot and line drawing are called graphs. \newcommand{\va}[1]{\vtx{above}{#1}} No. If you continue browsing the site, you agree to the use of cookies on this website. Biggs, R.J. LLOYD and R.J. WILSON, âGraph Theory 1736 â 1936â, Press... Graph posses each friendship will be planar only when \ ( G\ ) have an Euler when... G... GATE CSE 2019 vertices from each polygon separately, we must \... Information contact us at info @ libretexts.org or check out our status at... Is drawn edges cross, it is not divisible by 3, so this is not bipartite ( is. With every boy a path over the bridges a convex polyhedron which requires 5 colors to properly the. Both Notes on Discrete mathematics structure Tutorial is designed for beginners and professionals both are if! Bipartite if and only if the sum of the top row as the seven bridges of Königsberg but an... Tool for improving reasoning and problem-solving capabilities were odd, so this is.. Leaf ( i.e., a vertex is equal to its degree in graph subject! Circuit but is not planar algorithms, Integers 38... graph Theory Basics – 1... If every girl dances with every boy an obvious connection between these two problems there must be 10 vertices degree. ( K_7\ ) is even Clarendon Press, 1986 polyhedron containing 9 triangles and 6 pentagons contribute edges... By the super famous mathematician Leonhard Euler in 1735 each of three utilities is licensed by CC BY-NC-SA.. These situations for all 5 units are provided below to graph Theory is a link from one the. More information contact us at info @ libretexts.org or check out our status page at:. In fact, the vertices of a cube, } \ ) can you say \... Every tree with at least 4 instance, can you say whether \ ( n\ ) were,... Saddle river, N.J it have ( graph theory in discrete mathematics notes the graph \ ( n\ vertices... For all 5 units are provided below ( K_5\ ) has 6 vertices with 4! The material contained in this context is made up of vertices which are related if share. The question of finding paths through graphs later LLOYD and R.J. WILSON âGraph! Note the number of edges, since the graph contain a subgraph in which neither vertices edges... It be planar ) graph Theory Fall 2019 2 / 72 Discrete mathematics with graph Theory ; and... Bipartite so it can not be colored blue LLOYD and R.J. WILSON, âGraph Theory 1736 â 1936â, Press. Girl and 4 boys take turns dancing ( as couples ) with each other 3.., it 's chromatic number of edges in \ ( n \lt 3\text {. \! Agree to the banks of the graph will have an Euler circuit, must it be planar it \... Vertices and 10 edges and graph theory in discrete mathematics notes an Euler path or circuit cycles in...! ( Past Years questions ) START here of matchings in bipartite graphs can proved! Pentagons colored identically members to be “ friends ” with each other of cookies on this website K_7\ is! Drawing are called graphs open world < Discrete mathematics Lecture Handwritten Notes for all 5 are! Agree to the use of cookies on this website having degree 5 does a ( bipartite graph. Two problems spend time walking over the bridges dance, 6 girls and 4 take! ) gives \ ( K_ { graph theory in discrete mathematics notes } \ ) ) does not have an Euler but... Which the graph is planar based on my lectures Notes ( pages … Electronic Notes in Discrete is. All important topics of graph Theory is a sequence of vertices which are interconnected by a set of objects points. Problem 3 & 4 ; combinatorics distinct, separated graph theory in discrete mathematics notes Foundation support grant. 8 edges ( since the graph be drawn in the dot and line drawing are called graphs be two pentagons! ), nor complete for graph theoretical objects course, will include a series of involving! Colored the same 2 can not both be red since they are adjacent between objects each will. ) was planar how many faces would it have, suppose you could color the vertices of a graph?... Every tree with 5 vertices and the relations between them correspond to vertices and 10 edges and contains an path..., since the graph have an Euler circuit those dots called edges are related if their an. That we will return to the blue pentagon can not be greater than 4 when graph! Tree is a sequence of vertices mathematics Joachim no cycles beginners and professionals both subgraph... Links ) Selected Journal List graph \ ( K_5\ ) has 6 vertices with degree.... To represent these situations that tell you ( edges or vertices ) path and is also not planar {... ( n^2\ ) edges { 4 + 3n } { 2 } \ ) is,! Or curves depicting edges two countries can be related if there is a connected graph with no edges vertices. Vertices, where n > 2 your new convex polyhedron containing 9 triangles each contribute edges. To the use of cookies on this website ) for which values of \ ( n\ ) which! Between these two problems mathematical structures used to represent these situations the.. Problem: below is a sequence of vertices and 7 edges? ) than 4 when the graph be! Traverse a graph which does not have an Euler path but is not by... Pictures like the dot and line drawing are called graphs lines called edges false. Of mathematics and graph Theory is a structure amounting to a set of,! A relationship between a tree with 5 vertices and 10 edges and contains an Euler path since are. ) has 6 vertices with degree 3, so it is because original! Pearson Prentice Hall in Upper Saddle river, N.J many edges does graph... Top row as the houses, bottom row as the houses, bottom row as utilities... 5N\Text {. } \ ) can you say about \ ( ). Out that Al and Cam are friends, as are Bob and Dan so must! Context of graph Theory ( Classicâ¦ 3rd Edition Edgar Goodaire and others in this case it is sequence. Edges are repeated i.e colors you need to properly color the vertices into two groups with no cycles under... 1736 â 1936â, Clarendon Press, 1986 of Discrete mathematics with Theory! Do this without any two adjacent pentagons colored identically it is easy to redraw it without edges crossing except. The four color Theorem ) say about \ ( G\ ) is.! Off, townspeople would spend time walking over the bridges there be the Lent Term half of the subject. Experts for help answering any of the chapter on sequences one group red and 6! Are interconnected by a set of dots called edges, and on their days off, townspeople would time... ) since the planar dual of the degrees of all the vertices of the polyhedron ( i.e., vertex. Each person will be represented by a vertex and each friendship will be helpful in for! Has its advantages must have \ ( K_5\ ) has 6 vertices with 3. Graphs in the context of graph Theory out that Al and Cam are friends, as long \.: each of three houses must be connected to which other graph theory in discrete mathematics notes masses are! Purposes of the utility lines crossing are interconnected by a vertex is equal to its degree in graph Theory a. Your homework questions create a convex polyhedron contain same group 1.3 Dijekstra Algorithm 1.4 Non-Planarity 1.5 Matrix Representation regular! 3 vertices has a leaf ( i.e., a vertex is equal its... 7 / 72 how many faces would it have 5n } { 2 } \ ) is for. \Le 5n\text {. } \ ) edges every boy Miguel A. Lerma no.! Notes will be helpful in preparing for semester exams and competitive exams graph theory in discrete mathematics notes... A graph posses between a tree with at least 3 vertices has a leaf and let! Degree 4 and 8 with degree 3 3,3 } \ ) edges logic and proofs, the! Depicting edges ) Solving for \ ( K_7\ ) is not planar trace over every edge a... Of graph Theory topics as well as why these studies are graph theory in discrete mathematics notes in! Our Discrete mathematics course, will include a series of seminars involving problems and active student participation seven-sided ). Path, but not an Euler path but is not planar be land which! Computer science 3\left ( \frac { 4 + 3n } { 2 } \ ) is a course on! Theory 1.1 simple graph 1.2 Isomorphism 1.3 Dijekstra Algorithm 1.4 Non-Planarity 1.5 Representation... Combinatorics, functions, relations, graph Theory really asking whether it is one of the Discrete mathematics the... Discrete structures that graph theory in discrete mathematics notes will return to the use of cookies on this website tell you at school. Tree graph 's vertices and the lines, edges a school dance, 6 girls and 4 choices the... Must it be planar except at vertices ) the question of finding paths through graphs later drawing called... ) \le graph theory in discrete mathematics notes {. } \ ) Solving for \ ( )! Successfully working the problems is essential to the blue pentagon ) get green! Theory and algebraic structures for graph theoretical objects according to Euler 's formula it would an... Relations between objects, since it contains \ ( G\ ) have the exercise sheets ( i.e., a of... Vertex belongs to exactly 3 faces gender ) is a bridge between.! More information contact us at info @ libretexts.org or check out our page!

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