By Alan Tucker
This is often Alan Tuckers textbook on combinatorics and graph conception
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Extra resources for Applied Combinatorics (6th Edition)
18. 4, find all sets of three vertices that are adjacent to all the other vertices. Give a careful logical analysis to justify your answer. 19. 4 is altered by adding edges ( f, g), (g, j) and deleting (b, f ). 20. Repeat Example 4 for the edge cover and minimal corner surveillance when the network is formed by a regular array of north–south and east–west streets of size: (a) 3 streets by 3 streets (b) 4 streets by 4 streets (c) 5 streets by 5 streets 21. (a) A queen dominates any square on a chessboard in the same row, column, or diagonal as the queen.
14. Prove in a range graph that if a vertex (PL , PR ) consists of a valley and a peak, it will have degree 0. 15. Determine whether the following graphs are bipartite. 16b. 12 16. Determine whether the following graphs are bipartite. 16b. a a f c h m d b g d f g k j e e n i b h c 17. Suppose x and y are the only two vertices of odd degree in graph G, and x and y are not adjacent to each other. Show that G is connected if and only if the graph obtained from G by adding edge (x, y) is connected.
Exercise 10 involves duality. Exercises 15–26 build on Euler’s formula and the corollary e ≤ 3v − 6. The other exercises introduce new concepts. P1: PBU/OVY P2: PBU/OVY WB00623-01 WB00623-Tucker 40 QC: PBU/OVY T1: PBU October 25, 2011 13:59 Chapter 1 Elements of Graph Theory 1. 18b 2. Show that K 5 is nonplanar by the method in Example 2. 3. Which of the following graphs are planar? Find K 3,3 or K 5 configurations in the nonplanar graphs (almost all are K 3,3 ). (a) (b) a f d (c) c b a e f j (e) g e g (j) b e c (k) a e c d b (l) a c f e d c b f d a b d f f d e a d b c a (h) c h e d a i g c e (g) a b g b f f g (f) f e h h i f a g d a e b (i) c i c a (d) b d g b d h f e i h e g c f j d g b h c 4.
Applied Combinatorics (6th Edition) by Alan Tucker