This document summarizes an article about graphs and algorithms for finding trails in graphs. It discusses Eulerian graphs, which have a Eulerian circuit that visits every edge exactly once, and describes Fleury's algorithm for finding such a circuit. It also discusses Hamilton graphs, which have a Hamilton cycle that visits every vertex exactly once. Finally, it introduces weighted graphs, where edges have numeric weights, and explains that an Eulerian circuit is an optimal trail in a weighted Eulerian graph.

1. Math 178
Prof. 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

2. 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.

3. 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.