please send handwritten solution for Q5 for getting upvote 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. 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.

Let us make two claims and then prove it with reasoning: Claims:

Figure 2: The three faces are labeled - including the third 'outer face'. If there are no areas enclosed by edges (as in a 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 (such that 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 until no more edges can be added. Show / argue that in any planar triangulation, every face is a triangle (hence the name), i.e., that every face is surrounded by 3 edges.

Figure 2: The three faces are labeled - including the third 'outer face'. If there are no areas enclosed by edges (as in a 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 (such that 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 until no more edges can be added. Show / argue that in any planar triangulation, every face is a triangle (hence the name), i.e., that every face is surrounded by 3 edges.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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