please send handwritten solution for Q3 Problem 1: ColorfulConsider a planar graph - one that can be drawn in the plane without any edge crossings: We can consider not onlyFigure 1: Small Planar Graph, and Friendthe edges and vertices of such a graph, but also the faces - areas that are completely enclosed by edges.Figure 2: The three faces are labeled - including the third 'outer face'. If there are no areas enclosed by edges (as ina tree) we say there is 1 face. Figure 2: The three faces are labeled - including the third 'outer face'. If there are no areas enclosed by edges (as ina tree) we say there is 1 face.1) Show that for any planar graph, v – e + f = 2. (Note that in the above example, we have v = 6, e = 7, f = 3,and 6 – 7+3 = 2). As a hint, considering inducting on the number of edges. What does adding an edge (suchthat the graph is still planar) do to the number of faces?2) A planar triangulation is constructed from a planar graph, and adding edges without edge intersections untilno more edges can be added. Show / argue that in any planar triangulation, every face is a triangle (hence thename), i.e., that every face is surrounded by 3 edges.Figure 3: Orange edges showing two different triangulations of the graph - Note, the lower right version is actuallyincomplete and one more edge could be added. Where?3) What is the relationship between the number of faces and the number of edges in a triangulation? Justify.4) Show that the average degree of a vertex in the triangulation is strictly less than 6.5) Prove, by induction, that any planar graph can be colored in no more than 6 colors where no two verticesconnected by an edge share the same color.

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Answered: 2) A planar triangulation is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 8. A simple graph is called regular if every vertex of this graph has the same degree. How many vertices does a regular graph of degree 4 with 10 edges have? Justify your answer.
subject: discrete structures there may be several correct answers How many edges in the union of the graphs in Fig.1 and Fig.5?
Problem 7. Determine if a graph with the property specified below exists. If it exists, draw the graph and give its adjacency matrix; otherwise, prove that the graph doesn't exist. (a) A simple digraph with in-degrees 0, 1, 2, 2 and out-degrees 0, 1, 1, 3. (b) A simple digraph with in-degrees 0, 1, 1, 2 and out-degrees 0, 1, 1, 1.
subject: discrete structures there may be several correct answers How many edges in the subgraph induced by the set {?, ?, ?} of the graph in Fig.2?
Indicate if each of the two graphs are equal. Figure 1: Left: An undirected graph has 5 vertices. The vertices are arranged in the form of an inverted pentagon. From the top left vertex, moving clockwise, the vertices are labeled: a, b, c, d, and e. Undirected edges, line segments, are between the following vertices: a and b; a and c; b and c; c and d; e and d; and e and c. Figure 2: Right: The adjacency list representation of a graph. The list shows all the vertices, a through e, in a column from top to bottom. The adjacent vertices for each vertex in the column are placed in a row to the right of the corresponding vertex’s cell in the column. An arrow points from each cell in the column to its corresponding row on the right. Data from the list, as follows: Vertex a is adjacent to vertices b and c. Vertex b is adjacent to vertices a and c. Vertex c is adjacent to vertices a, b, d, and e. Vertex d is adjacent to vertices c and e. Vertex e is adjacent to vertices c and d.
I know that there is an answered question on the question bank, but there are syntax errors the computer caused, so I don't understand it correctly. Prove that the two graphs below are isomorphic. Figure 4: Two undirected graphs. Each graph has 6 vertices. The vertices in the ﬁrst graph are arranged in two rows and 3 columns. From left to right, the vertices in the top row are 1, 2, and 3. From left to right, the vertices in the bottom row are 6, 5, and 4. Undirected edges, line segments, are between the following vertices: 1 and 2; 2 and 3; 1 and 5; 2 and 5; 5 and 3; 2 and 4; 3 and 6; 6 and 5; and 5 and 4. The vertices in the second graph are a through f. Vertices d, a, and c, are vertically inline. Vertices e, f, and b, are horizontally to the right of vertices d, a, and c, respectively. Undirected edges, line segments, are between the following vertices: a and d; a and c; a and e; a and b; d and b; a and f; e and f; c and f; and b and f.
If a graph has vertices of degrees 1, 1, 2, 3, and 3, how many edges does it have? Why?
Problem 3 with A-H graph sketches ??
Figure 1 below shows three different graphs. The graph in Fig. 1(a) is connected; the graph in Fig. 1(b) is disconnected and has two sections; the graph in Fig. 1(c) is disconnected and has three sections (the isolated vertex G is a section— that’s as small a section as you can get!). Notice that the only difference between the disconnected graph in Fig. 1(b) and the connected graph in Fig. 1(a) is the edge BF. Think of BF as a “bridge” that connects the two sections of the graph in Fig. 1(b). Not surprisingly, we call such an edge a bridge. A bridge in a connected graph is an edge that keeps the graph connected—if the bridge were not there, the graph would be disconnected. The graph in Fig. 1(a) has three bridges: BF, FG, and FH Draw the graphs for item (c)and then, list all the bridges in each of the three graphs graphs: c) the graph with vertex set {A, B, C, D, E} and edge list AB, BC, CD, DE
Problem 1. Let A= [−2,2] ×[−2,2] and B= [−1,1] ×[−1,1]. Graph A, B,A∪B, A∩B, and A∆B. (ATTN: All the sets above are subsets of R2 i.e. the xOyplane).
(a) A schedule is to be made with ﬁve football teams. Each team is to play two other teams. Explain how to make a graph model of this problem. (b) Show that except for interchanging names of teams, there is only one possible graph in part
Question 10 . Count the number V of vertices, the number E of edges and the number R of regions of each map, also check compliance with Euler's formula.

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