MAT 2720 Discrete Mathematics Section 8 2 Paths

MAT 2720 Discrete Mathematics Section 8. 2 Paths and Cycles http: //myhome. spu. edu/lauw

Goals l Paths and Cycles • Definitions and Examples • They are the “alphabets” of graph modeling

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.

Connected Graph

Example 2 The graph is not connected because …

Subgraph

Subgraph

Subgraph

Example 3 – 2 min. Tryout How many subgraphs are there with 3 edges?

Components

Components

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

Simple Paths

Cycles

Simple Cycles

Degree l

Degree l

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

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

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

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

Theorems 8. 2. 17 & 8. 2. 18: G has an Euler cycle if and only if G is connected and every vertex has even degree.

Theorems 8. 2. 17 & 8. 2. 18: 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 8. 2. 21

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

Corollary 8. 2. 22

2017 Stop Here!

Theorem 8. 2. 23

Theorem 8. 2. 24