euler paths and circuits

Euler Path

A path that passes through each edge of a graph exactly one time is called​ a(n) ______ path.

A connected graph has even vertices​ A, B, and​ C, and odd​ vertices, D and E. Each Euler path must begin at vertex D and end at vertex​ _______, or begin at vertex​ _______ and end at vertex​ _______.

E E D.

Euler Theorem

is used to determine if a graph contains euler paths or euler circuits

Euler Circuit

A circuit that travels through every edge of a graph exactly once is called​ a/an _______ circuit.

two

A connected graph has at least one Euler​ path, but no Euler​ circuit, if the graph has exactly​ _______ odd​ vertices/vertex.

​Euler's Theorem enables us to count a​ graph's odd vertices and determine if it has an Euler path or an Euler circuit. A procedure for finding such paths and circuits is called​ _______ Algorithm. When using this algorithm and faced with a choice of edges to​ trace, choose an edge that is not​ a/an _______. Travel over such an edge only if there is no alternative.

Fleury's Algorithm. bridge.

Every Euler path is an Euler circuit.

The statement is false because both an Euler circuit and an Euler path are paths that travel through every edge of a graph once and only once. An Euler circuit also begins and ends on the same vertex. An Euler path does not have to begin and end on the same vertex.

​Euler's Theorem provides a procedure for finding Euler paths and Euler circuits.

The statement is false. While​ Euler's Theorem provides a way to determine whether or not a graph is an Euler path or an Euler​ circuit, it does not provide a means for finding an Euler path or an Euler circuit within a graph

Every Euler circuit is an Euler path.

The statement is true because both an Euler circuit and an Euler path are paths that travel through every edge of a graph once and only once. An Euler circuit also begins and ends on the same vertex.

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

odd