Graph Theory

Applied Combinatorics (6th Edition) by Alan Tucker

By Alan Tucker

This can be Alan Tuckers textbook on combinatorics and graph concept

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Extra info for Applied Combinatorics (6th Edition)

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1 Graph Models 13 (b) Use your result in part (a) to produce a minimal edge cover in these graphs. 24. What is the largest independent set in a circuit of length 7? Of length n? 25. (a) What is the largest independent set possible in a connected seven-vertex graph? Draw the graph. (b) What is the largest independent set possible in a seven-vertex graph (need not be connected)? Draw the graph. 26. , edge (a, e) is directed from a to e] 27. Show that the vertex basis in a directed graph is unique if there is no sequence of directed edges that forms a circuit in the graph.

13b, the answer is obviously yes. We claim that vertices (A, Z ) and (M, M) in any range graph have degree 1, whereas every other vertex in the range graph has degree 2 or 4. (A, Z ) has degree 1 because when both people start climbing up the range from their respective sides, they have no choice initially but to climb upward until one arrives at a peak. 13a, the first peak encountered is C on the left, and so the one edge from (A, Z ) goes to (C, X). A similar argument applies at (M, M). Next consider a vertex (PL , PR ) where one point is a peak and the other point is neither peak nor valley, such as (E, W).

11. (a) Show that if a circuit in a planar graph encloses exactly two regions, each of which has an even number of boundary edges, then the circuit has even length. P1: PBU/OVY P2: PBU/OVY WB00623-01 WB00623-Tucker 42 QC: PBU/OVY October 25, 2011 T1: PBU 13:59 Chapter 1 Elements of Graph Theory (b) Show that if a circuit in a planar graph encloses a collection of regions, each of which has an even number of boundary edges, then the circuit has even length. 12. The crossing number c(G) of a graph G is the minimum number of pairs of crossing edges in a depiction of G.

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