Graph Theory

Abstract

Graphs are introduced using binary relations. Complete graphs, complete bipartite graphs, and complements are defined, as are connectedness and degree. The second section is devoted to the Königsberg Bridge problem and traversability. Then general walks are introduced, together with paths and cycles.

Sections 7.4, 7.5, and 7.6 treat three important applications: shortest paths, minimal spanning trees, and Hamilton circuits. Finally, a section is devoted to the enumeration of Hamilton cycles and the Traveling Salesman problem. It is not known whether there is a polynomial-time solution to this problem, so in addition to brute-force enumeration, two approximation algorithms—the nearest-neighbor algorithm and the sorted-edges algorithm—are outlined.