By John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (auth.)

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Unfortunately, the townspeople of Verona just had to learn to deal with the feuding families, for K 3 . 3 is nonplanar, and we will see an explanation shortly. What is involved in showing that a graph G is nonplanar? In theory, one would have to show that every possible drawing of G is not a planar representation. Since considering every individual drawing is out of the question, we need some other tools. Given a planar representation of a graph G, a region is a maximal section of the plane in which any two points can be joined by a curve that does not intersect any part of G.

If not, repeat step ii. 34 demonstrates Kruskal's algorithm applied to the city graph. The minimum weight is 210. 2 Trees 23 It is certainly possible for different trees to result from two different applications of Kruskal's algorithm. For instance, in the second step we could have chosen the edge between Marion and Lenoir instead of the one that was chosen. Even so, the total weight of resulting trees is the same, and each such tree is a minimum weight spanning tree. It should be clear from the algorithm itself that the subgraph built is in fact a spanning tree of G.

8. Let G be a graph of order n. :s xCG) + xCG). a. n b. 3 The Four Color Problem That doesn't sound too hard. -Star Wars The Four Color Problem Is it true that the countries on any given map can be colored with four or fewer colors in such a way that adjacent countries are colored differently? The seemingly simple Four Color Problem was introduced in 1852 by Francis Guthrie, a student of Augustus DeMorgan. The first written reference to the problem is a letter from DeMorgan to Sir William Rowan Hamilton.