## Graph Theory Lecture Notes 11

### Applications of Eulerian Chains and Paths

#### Chinese Postman Problem

Given a graph, where the edges are considered to be streets that the postman must traverse, how can
a walk be arranged so that every edge is traversed and the postman begins and ends at the post
office?
The problem calls for an eulerian circuit, which may be impossible if there are vertices of odd
degree. However, if we add some edges between already adjacent vertices (corresponding to the
postman walking over a street another time), we can arrange to have all vertices of even degree. We
would like to do this by adding as few edges as possible.

#### Street Sweeping Problem

Similar to the Chinese Postman Problem, but now the edges have numbers associated with them,
which indicate how long it takes to traverse the edge without sweeping (deadheading). Instead of
minimizing the number of deadheading edges as in the postman problem, we now want to minimize
the sum of these deadheading times.

#### De Bruijn Sequences

Given an alphabet of p letters, and words of length n made from these letters, a (p,n)- De Bruijn
sequence is a circular sequence of letters of the alphabet, such that each n length word is realized
once in consecutive terms of the sequence. Such a sequence must have size p^{n}.
A De Bruijn diagram (really a digraph), has as its vertices all words of length n-1. The arcs join
vertices of the form b_{1}b_{2} ... b_{n-1} to b_{2}b_{3} ... b_{n}. They are labeled with the word b_{1} ... b_{n}.

This digraph is weakly connected, and every vertex has in-degree = out-degree = p.

A (p, n)- De Bruijn sequence is given by the first letter of each label on an eulerian closed path.