Chapter 14.1 & 14.2

Euler's Theorem for Euler Circuits

1. If a graph is connected and 2. all vertices are even, then the graph has Euler circuit.

A path that passes through each edge of a graph exactly one time is called an ____________ path.

Euler

An _________ circuit is a circuit that travels through every edge of a graph once and only once.

Euler

At least one ___________ path exists in any connected graph with either no or exactly two odd vertices.

Euler

disconnected graph

a graph that is not connected; a graph where it is not possible to reach at least one vertex from any starting vertex by traversing edges. For example, in the graph below you cannot go from vertex A to vertex F by traversing the edges.

connected graph

a graph where it is possible to reach any vertex from any specified starting vertex by traversing edges. The graph below is an example of a connected graph.

Euler path

a path that travels through every edge of a graph once and only once. Each edge must be traveled and no edge can be retraced.

vertex

a point on a graph where one or more lines (called edges) end. The plural is vertices.

The number of edges that connect to a vertex is called the ___________ of the vertex.

degree

The degree of a ___________ is the number of edges that connect to that vertex.

edges

Two graphs are _____________ if they have the same number of vertices connected to each other in the same way.

equivalent - The placement of the vertices and the shapes of the edges are unimportant.

For an Euler​ path, every edge of the graph must be traveled ________ once.

exactly

If a graph has an Euler​ path, it is said that the __________ is traversable.

graph

In a​ _______________, the important information is which vertices are connected by edges.

graph

edge

a line or link joining two vertices on a graph.

graph

a mathematical model using a finite set of points (called vertices) and lines segments (called edges) that start and end at vertices to describe a real-world situation.

circuit

a path that starts and ends at the same vertex.

loop

an edge that starts and ends at the same vertex.

A connected graph has 62 even vertices and no odd vertices. Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

1. an Euler circuit. 2. no odd vertices.

Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

1. an Euler path​ (but not an Euler​ circuit). 2. an Euler path​ (but not an Euler​ circuit).

A _______________ graph has at least one Euler​ path, but no Euler​ circuit, if the graph has exactly ________ odd vertices.

1. connected 2. two

A connected graph has 92 even vertices and eight odd vertices. Determine whether the graph has an Euler path​ (but not an Euler​ circuit), an Euler​ circuit, or neither an Euler path nor an Euler​ circuit, and explain why.

1. neither an Euler path nor an Euler circuit. 2. neither an Euler path nor an Euler circuit.

A connected graph has at least one Euler​ circuit, which, by​ definition, is also an Euler​ path, if the graph has _______ odd vertices.

1. no

Euler's Theorem for Euler Paths

If a graph is 1. connected and 2. has exactly two odd vertices then the graph has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other.

T or F....A graph can be used to model friendships between the coworkers at a company.

True

Euler circuit

a circuit (starts and ends at the same vertex) that covers every edge once and only once.

path

a connected sequence of edges on a graph starting at a vertex and ending at a vertex, usually described by naming the vertices visited in traversing it. According to our textbook, a path cannot repeat an edge.

Two vertices in a graph are said to be ___________ vertices if there is at least one edge connecting them.

adjacent

bridge

an edge (ONE edge) that if you remove it from a connected graph you end up with a disconnected graph. The graph below has two bridges. Can you identify them? The two bridges are edge AB and edge BE. If you remove either of those edges the graph becomes disconnected. Students sometimes get confused about how an edge like AB will disconnect the graph. If you think of the vertices as cities and the edges as connecting links between the cities, then the removal of edge AB, isolates (or disconnects) city A from the rest of the cities.

A ____________ is an edge that if removed from a connected graph would leave behind a disconnected graph.

bridge

If an edge is removed from a connected graph and leaves behind a disconnected​ graph, such an edge is called a ____________.

bridge

A ___________ is a path that begins and ends at the same vertex.

circuit

A _____________ is a path that begins and ends at the same vertex.​

circuit

Like all​ _________, an Euler circuit must begin and end at the same vertex.

circuits

A connected graph has no Euler paths and no Euler circuits if the graph has more than two _________ vertices.

odd

Each Euler path must start at one of the __________ vertices and end at the other one.

odd

A ________ in a graph is a sequence of adjacent vertices and the edges connecting them.

path

A ___________ in a graph is a sequence of adjacent vertices and the edges connecting them

path

Not every ______________ ends at the same vertex where it​ starts, not every path is a circuit.

path

even-vertex

the degree of the vertex is an even number.

odd-vertex

the degree of the vertex is an odd number.

degree of a vertex

the number of edges at that particular vertex.

equivalent graphs

two graphs are equivalent if they have the same number of vertices connected to each other in the same way. The placement of the vertices and the shapes of the edges are unimportant.

adjacent vertices

two vertices that have at least one edge connecting them.

An Euler circuit can start and end at any __________ .

vertex

If a loop connects a __________ to​ itself, that loop contributes 2 to the degree of the vertex.

vertex

If a graph has more than two odd​ ___________ , then it has no Euler paths and no Euler circuits.

vertices