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4) Show that the average degree of a vertex in the triangulation is strictly less than 6.

4) Show that the average degree of a vertex in the triangulation is strictly less than 6.

Advanced Engineering Mathematics
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 1: Colorful Consider a planar graph - one that can be drawn in the plane without any edge crossings: We can consider not only Figure 1: Small Planar Graph, and Friend the 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 in a tree) we say there is 1 face.
Transcribed Image Text:Problem 1: Colorful Consider a planar graph - one that can be drawn in the plane without any edge crossings: We can consider not only Figure 1: Small Planar Graph, and Friend the 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 in a 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 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 3: Orange edges showing two different triangulations of the graph - Note, the lower right version is actually incomplete 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 vertices connected by an edge share the same color.
Transcribed Image Text: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 3: Orange edges showing two different triangulations of the graph - Note, the lower right version is actually incomplete 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 vertices connected by an edge share the same color.
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