A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

The mixed postman problem consists of finding a minimum cost tour of a mixed graph M = (V, E, A) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The extra cost of a mixed postman tour T is the cost of T minus the cost of the edges and arcs of M . We prove that it is to approximate the minimum extra cost of a mixed postman tour. A related

Probably the oldest and best known of all problems in graph theory centers on the bridges over the river Pregel in the city of Ko.. nigsberg (presently called Kaliningrad in Russia). The legend says that the inhabitants of Ko.. nigsberg amused themselves by trying to determine a route across each of the bridges between the two islands (A and B in Figure 5.1.1), both river banks (C and D of Figure 5.1.1) and back to their starting point using each bridge exactly one time. After many attempts, they all came to believe that such a route was not possible. In 1736, Leonhard Euler [15] published what is believed to be the first paper on graph theory, in which he investigated the Ko.. nigsberg bridge problem in mathematical terms.