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By W. D. Wallis

Concisely written, mild advent to graph idea compatible as a textbook or for self-study Graph-theoretic functions from assorted fields (computer technological know-how, engineering, chemistry, administration technology) second ed. comprises new chapters on labeling and communications networks and small worlds, in addition to improved beginner's fabric Many extra adjustments, advancements, and corrections caused by school room use

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SO S5 = e, 55 = {s, a, e, b, d, e} and fee) = 10. e has predecessor d. From now on b need not be considered. The nearest vertex to e is f; wee, f) = 5 so £(e) + wee, f) = 11. The nearest vertex to d is g; wed, g) = 3 so fed) + wed, g) = 11. The nearest vertex to e is h; wee, h) = 2 so fee) + wee, h) = 12. Either f or g could be chosen. For convenience, suppose the earlier member of the alphabet is always chosen when equal £-values occur. Then S6 = f, 56 = {s, a, e, b, d, e, f} and £(f) = 11. f has predecessor e.

However, no good necessary and sufficient conditions are known for the existence of Hamilton cycles. The following result is a useful sufficient condition. 7. If G is a graph with v vertices. v ::: 3. and d(x) x and yare nonadjacent vertices of G, then G is Hamiltonian. 5 Hamilton Cycles 35 Proof. Suppose the theorem is false. Choose a v such that there is a v-vertex counterexample, and select a graph G on v vertices that has the maximum number of edges among counterexamples. Choose two nonadjacent vertices p and q: because of the maximality of G, G + pq must be Hamiltonian.

Iii) Find a solution for 9 people and 4 days. 3, later, for a general solution. 6 The Traveling Salesman Problem Suppose a traveling saleman wishes to visit several cities. If the cities are represented as vertices and the possible routes between them as edges, then the salesman's itinerary is a Hamilton cycle in the graph. In most cases one can associate a cost with every edge. Depending on the salesman's priorities, the cost might be a dollar cost such as airfare, a number of miles, or a number of hours.

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