MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles

Goals Paths and Cycles Definitions and Examples More Definitions

Definitions

Example 1 (a) Write down a path from b to e with length 4.

Example 1 (b) Write down a path from b to e with length 5.

Example 1 (c) Write down a path from b to e with length 6.

Definitions

Example 2 The graph is not connected because …

Definitions

Example 3 How many subgraphs are there with 3 edges?

Definitions

Connected Graph & Component What can we say about the components of a graph if it is connected?

Connected Graph & Component What can we say about the graph if it has exactly one component?

Theorem A graph is connected if and only if it has exactly one component

Definitions

The degree of a vertex v, denoted by  (v), is the number of edges incident on v

Definitions The degree of a vertex v, denoted by  (v), is the number of edges incident on v

The Königsberg bridge problem Euler (1736) Is it possible to cross all seven bridges just once and return to the starting point?

The Königsberg bridge problem Edges represent bridges and each vertex represents a region.

The Königsberg bridge problem Euler (1736) Is it possible to find a cycle that includes all the edges and vertices of the graph?

Definitions An Euler cycle is a cycle that includes all the edges and vertices of the graph

Theorems & : G has an Euler cycle if and only if G is connected and every vertex has even degree.

Theorems & : G has an Euler cycle if and only if G is connected and every vertex has even degree.

Example 4(a) Determine if the graph has an Euler cycle.

Example 4(b) Find an Euler cycle.

Observation The sum of the degrees of all the vertices is even.

Example 5 (a) What is the sum of the degrees of all the vertices?

Example 5 (b) What is the number of edges?

Example 5 (c) What is the relationship and why?

Theorem

Example 6 Is it possible to draw a graph with 6 vertices and degrees 1,1,2,2,2,3?

Corollary

Theorem

Theorem