Math 160, Chapter 5, Graphs, Euler Circuits
Definition 1.1. a) A graph is a finite set of vertices V = {A, B, C, D, E} together
with a set of edges E = {AB, AC, BC, DD, DE}, connecting the vertices. Each
edge connects two vertices.
b) A loop is an edge that connects a vertex to itself.
c) A path is a connected sequence of edges starting at one vertex and ending at
another (possibly the same). No edge can appear more than once on a path. (A
vertex may be visited more than once on a path.)
d) A circuit is a path that starts and ends at the same vertex. A reference
point for a circuit is a designated starting point.
Definition 1.2. a) A connected graph is a graph such that any two vertices can
be joined by a path.
b) An edge of a connected graph is called a bridge if its removal causes the
graph to be disconnected.
Definition 1.3. a) An Euler path is a path that covers (contains) every edge of
the graph exactly once.
b) An Euler circuit is a circuit that covers (contains) every edge of the graph
exactly once.
Definition 1.4. The degree of a vertex is the number of edges meeting at the
vertex. A loop counts as two edges meeting at the vertex.
Theorem 1.1. Euler’s Theorem. Let G be a connected graph. Then
a) If some vertex has odd degree, then G has no Euler circuit.
b) If every vertex has even degree, then G has an Euler circuit.
c) If there are exactly two vertices of odd degree, then G has an Euler path that
starts at one of these vertices and ends at the other.
d) If there are more than two vertices of odd degree, then G has no Euler path.
Definition 1.5. An algorithm is a step by step procedure for carrying out a given
task.
Fleury’s Algorithm for constructing an Euler circuit.
1. First check that every vertex has even degree. If not, there is no Euler circuit.
2. Start at any vertex A.
3. Proceed along any edge that is not a bridge between parts of the graph that
still need to be covered (that is, don’t burn your bridges behind you).
4. Keep repeating this process, and eventually you will end up back at A.
Fleury’s Algorithm for constructing an Euler path.
1. First identify the two vertices of odd degree, say A and B.
2. Start at A.
3. Proceed along any edge that is not a bridge between parts of the graph that
still need to be covered, if possible. If no such edge is left, then cross the bridge.
4. Keep repeating this process, and eventually you will end up at B.
Definition 1.6. To Eulerize a graph means to add edges so that the resulting
graph has an Euler circuit.
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Procedure for Eulerizing a graph.
1. Circle all vertices of odd degree.
2. Add edges that duplicate existing edges so that all vertices have even degree.
Finding a “circuit”∗ that minimizes duplication of edges. If an Euler circuit
exists then this is the optimal solution. Otherwise
1. Eulerize the graph, adding as few edges as possible.
2. Find an Euler circuit on the new Eulerized graph.
3. Now interpret the multiple edges connecting the same vertices as the same
edge repeated twice along the circuit.
* Note: We put quote marks here because this isn’t a true circuit. A true circuit
can cover each edge at most once.