{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lect_38 - Planar Graphs II Margaret M Fleck 30 April 2010...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Planar Graphs II Margaret M. Fleck 30 April 2010 This lecture continues the discussion of planar graphs (section 9.7 of Rosen). 1 Announcements Makeup quiz last day of classes (at the start of class). Your room for the final exam (Friday the 7th, 7-10pm) is based on the first letter of your last name: A-H Roger Adams Lab 116 I-W Business Instructional Facility (515 E. Gregory opposite Armory) 1001 X-Z Roger Adams Lab 116 The conflict exam (Monday the 10th, 1:30-4:30pm) is in 269 Everett Lab. 2 A corollary of Euler’s formula Suppose G is a connected simple planar graph, with v vertices, e edges, and f faces, where v 3. Then e 3 v - 6. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Proof: The sum of the degrees of the regions is equal to twice the number of edges. But each region must have degree 3. So we have 2 e 3 f . Then 2 3 e f . Euler’s formula says that v - e + f = 2, so f = e - v +2. Combining this with 2 3 e f , we get e - v + 2 2 3 e So e 3 - v + 2 0. So e 3 v - 2. Therefore e 3 v - 6. We can also use this formula to show that the graph K 5 isn’t planar. K 5 has five vertices and 10 edges. This isn’t consistent with the formula e 3 v - 6. Unfortunately, this trick doesn’t work for K 3 , 3 , which isn’t planar but satisfies the equation (with 6 vertices and 9 edges). 3 Another corollary In a similar way, we can show that if G is a connected planar simple graph with e edges and v vertices, with v 3, and if G has no circuits of length 3, then e 2 v - 4. Proof: The sum of the degrees of the regions is equal to twice the number of edges. But each region must have degree 4 because we have no circuits of length 3. So we have 2 e 4 f . Then 1 2 e f .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}