Math 178Prof. Bradley W. Jackson Kaya Ota 2/15/2015

UMAP Module 589 “Traversability in Graphs”

Mathematically, a graph consists a finite collection of points (i.e. vertices) and

a collection of line or curves (i.e. edges) that joins two vertices. There are a lot of

situations where we can model with a graph. For example, the most “efficient” way

for a postman whose job is to pick mails at a post office, deliver them, and return to

the post office. In that case, we consider shipping addresses as vertices and streets

the postman passes as edges. When we think about a network of something, a graph

can be a good model to make a problem simple. In this article, “Traversability in

Graphs” introduces two categories of graphs, and two algorithms to find a certain

trail in a given graph, and generalization of graphs.

The first category of graphs is called Eulerian graph. It is a graph that

contains a Eulerian circuit. Eulerian circuit is a trail that we are allowed to walk

through the same vertices but we cannot walk through the same edge. Leonhard

Euler make a wonderful observation about the existence of a Eulerian circuit in

general graph (so we call the graph Eulerian). His first observation is that a graph

needs to be connected, which means we can traverse from any vertex to any another

vertex in the graph. His second observation is that the degree of every vertex needs

to be even. So, we now know a graph G as a theory is Eulerian if and only if G is

connected and every vertex has even degree. This article shows the algorithm called

Fleury’s algorithm to find which trail is a Eulerian circuit in the graph. Fleury’s

algorithm is, first start from any vertex u of a given graph G, and each time we use an

edge, we delete the edge unless there is no alternatives. However, if the edge we use

is a bridge (i.e. an edge that makes a graph “connected”) then we do not erase the

edge. The reason why the article introduces this algorithm is Fleury’s algorithm is

considered to be good because it can implement as a program easily.

The second category of graph in this article is called Hamilton graphs. A

Hamilton graph contains at least one Hamilton cycle. Hamilton circuit is a trail

where we can walk through each vertex only once. Unlike a Eulerian graph,

Hamilton graph does not have a useful and observable characterization for its

existence. However, we can still derive a theorem by considering the completeness

of closure of a graph. The author of this article derives it as theorem, that is if G is a

graph with tree or more vertices and closure of the graph is complete, then G is a

Hamilton graph. The completeness of the closure of a graph is sufficient condition.

Finally, we can more generalize a graph to get more useful and real

mathematical model. A weighted graph is one of more generalized graph. It is a

graph where each edge is assigned a non-negative number called weight or cost of

the edge. So, the cost we need to pay when we pass a specific graph is no longer

same. Interestingly, if we know our given graph is Eulerian weighted graph, then

Eulerian circuit is optimal trail, which means minimum total weight in the given

graph which starting and ending vertex is identical and we walk through every

vertex and edges at least once. The reason why an Eulerian circuit is an optimal trail

is solved by Fleury’s algorithm.

In this article, we mainly discussed about Eulerian graph, Eulerian circuit

Hamilton graph, Hamilton cycle and weighted graph, and what they are.

We can simulate so many things with graph, not only a postman’s route but also

friendships on facebook, linedin. Thus, Graph theory is good to know to consider

about a collection of object in some specified manner.