Math 203
Eulerization – Why and How
An Euler circuit, when it exists, describes the most efficient solution to any problem where tasks have to
be done along the edges of a graph. Examples of such tasks abound if you try to think of them, for
example mail or newspaper delivery, garbage collection, parking meter monitoring, street sweeping, even
walking though a store’s aisles looking for unadvertised specials (or through a gift shop looking for
inspiration right before your significant other’s birthday)! But many of the graphs involved don’t have an
Euler path, or even if it does we may have to start at the wrong vertex (if you’re a mail carrier, you have to
start at the post office, even if that isn’t an odd vertex - you may have to end there too). However, if the
streets in your neighborhood don’t have an Euler circuit, is it okay with you if your garbage doesn’t get
collected? I (and your neighbors) hope not.
When the graph of edges we need to cover to perform an important task doesn’t have the Euler
path/circuit that meets our needs, what we must do is usually to cover paths not needed for the task and/or
to cover parts of our path twice. In the graph, that corresponds to adding edges to represent this extra
travel – the process of adding edges to get to a graph with an Euler path/circuit is called eulerizing. Any
edge we have to cover a second time becomes an edge that gets duplicated in the graph – covering the
edge and its duplicate really means traveling that edge twice. Any new path we take that wasn’t in the
graph becomes a new edge between the appropriate vertices on the old graph. (For example, in the
newspaper delivery application there might be a block where nobody subscribes, so that isn’t on our initial
diagram, but that block can be added to the graph if it helps form an efficient route.) However, most of
the time the edges already in the graph represent our options – adding a new edge between two vertices
that don’t already have an edge between them usually represents something like bulldozing everything
between two intersections and building a new direct road connecting them. Normally, that’s not feasible.
So to simplify the setting in dealing with these problems, we’ll limit ourselves to looking at changes in the
graph that duplicate existing edges. We’ll also assume for simplicity that all edges are equally costly to
travel a second time (costly may be in terms of time, money, or effort), implying that the most efficient
eulerization is the one that minimizes the number of edges we duplicate. (It’s not that hard to tweak the
process a little to fix things if either of these assumptions is false in a particular application.)
We’ll focus first on problems where we’d like to have an Euler circuit, i.e., we want to cover every edge
and end up at the vertex where we start. That means we need to add edges until every vertex is of even
degree. What happens when we duplicate an edge? Both vertices at the ends of the edge change in degree
parity (i.e. from odd to even or even to odd). If we have an odd degree vertex, and we duplicate an edge
to make that vertex even, if the other end of the edge was an odd vertex also, great, we’ve made both
