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2 "3 = 3. 6-J/7"3 , . "» is empty. 39. Building a labeled tree. Let G be a graph with vertices v1, v2 , ... , Vn. j• is defined by _! j - l 0 if v; and v j are adjacent, . otherwise. Let G be a graph with vertices v 1, v 2 , ... Vn. j. is defined by _! j- deg( v;) O if i = j, otherwise. 40. 40 has six vertices. Its degree matrix D and adjacency matrix A are 0 0 0 0 1 0 0 2 0 0 0 0 2 0 D= 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 3 0 1 0 0 0 0 0 3 0 A= 0 1 0 0 1 0 Let M be ann x n matrix.

While the Appel-Haken proof is accepted as being valid, mathematicians still search for alternative proofs. Robertson, Sanders, Seymour, and Thomas [125] have probably come the closest to finding a short and clever proof, but theirs still requires a number of computer calculations. In a 1998 article [139], Robin Thomas said the following. For the purposes of this survey, let me telescope the difficulties with the A&H proof into two points: ( 1) part of the proof uses a computer and cannot be verified by hand, and (2) even the part that is supposedly hand-checkable has not, as far as I know, been independently verified in its entirety....

7. Let G be of order n 2:: 11. Show that at least one of G and G is non planar. 8. 2) of a planar graph is less than six. 3 Planarity 39 9. 14 is not true. 10. Find a 4-regular planar graph, and prove that it is unique. 11. A planar graph G is called maximal planar if the addition of any edge to G creates a nonplanar graph. a. Show that every region of a maximal planar graph is a triangle. b. If maximal planar graph has order n, how many edges and regions does it have? 3 Regular Polyhedra We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.

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