# A8 - planar graph with at least two vertices cannot be...

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Due: Friday, July 23 Math 239 Assignment 8 1. Prove that the complement of the cube is not planar. 2. (a) Prove that a planar graph with girth at least six must have a vertex of degree at most two. (b) Using the previous part, show that a planar graph with girth at least six is 3-colorable. 3. By ﬁnding a subdivision of K 3,3 , prove that the 4-cube is not planar. 4. Find a connected cubic planar bipartite graph on 14 vertices. 5. A planar graph is self dual if it is isomorphic to its dual. Prove that a connected self dual
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Unformatted text preview: planar graph with at least two vertices cannot be bipartite. 6. If M and N are matchings in G and C is a cycle all of whose edges belong to M ∪ N , prove that C has even length. 7. Show that a tree has at most one perfect matching. 8. A subset S of the vertices of a graph is independent if no two vertices in it are adjacent. If G is regular with degree at least one and S is an independent set of vertices in G , prove that | S |≤ 1 2 | V ( G ) | ....
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## This note was uploaded on 11/26/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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