The Public Vehicle Routing Problem (PVRP) is to find the minimum total cost routes of M or less Public-Vehicles to traverse the required arcs(streets) at least once, and return to their starting depot on a directed network. In this paper, first, a mathematical model is formulated as minimal cost flow model with illegal subtour elimination constraints, and with the fixed cost and routing cost as an objective function. Second, an efficient branch and bound algorithm is developed to obtain an exact solution. A subproblem in this method is a minimal cost flow problem relaxing illegal subtour elimination constraints. The branching strategy is a variable dichotomy method according to the entering nonrequired arcs which are candidates to eneter into an illegal subtour. To accelerate the fathoming process, a tighter lower bound of a candidate subproblem is calculated by using a minimum reduced coast of the entering nonrequired arcs. Computational results based on randomly generated networks report that the developed algorithm is efficient.