Exam 5: The Mathematics of Getting Around: Euler Paths and Circuits

Your college campus has designed an underground tumnel system to connect five of the main buildings on campus. A map of the underground system is shown below. Suppose that you start in Meconi Hall, end in The Glasgow Center, and plan to travel through each and every tunnel, what is the fewest number of tunnels that you would need to travel through more than once in order to accomplish this task?

Consider the graph with vertex set and edge set What is the degree of vertex H ?

Consider the graph with vertex set ={A, B, C, D, E, F} and edge set = What is the degree of vertex E ?

Consider the graph with vertex set V = {A,,BC,X,Y,Z} and edge set E = {ABA,,CAY,AX,BYC,C,CZ}. Does this graph contain an Euler Path? Explain.

How many edges are bridges in the following graph?

Consider the graph with vertex set ={G, H, O, S, T} and edge set = {G H, G S, G S, G T, H O, O T, S T} . Which edges, if any, are bridges for the graph?

Tom is dancing at a local discothèque with his friends Matt, Bill, Lisa, and Allison. Tom dances with Lisa once and Allison twice. Bill dances with Lisa two times but does not dance with Allison, and Matt dances with Lisa and Allison one time each. Using people as vertices and dances as edges construct a graph which models this situation.

Consider the graph with vertex set ={A, B, C, D, E, F} and edge set = {AB, AB, AC, AE, AF, BD, BF, CD, CE, DF, EE, EF} . List all edges adjacent to edge D F .

-Consider the graph above ; what is the vertex set?

How many edges are bridges in the following grapgh ?

-Consider the graph above ; state the degree of vertex A.

Consider the graph with vertex set V = {A,,BC,X,Y,Z} and edge set E = {ABA,,CAY,AX,BYC,C,CZ}. Which edges, if any, are bridges for the graph?

For the graph shown below, which vertices are adjacent to vertex C?

Consider the graph with vertex set ={G, H, O, S, T} and edge set = {G H, G S, G S, G T, H O, O T, S T} . This graph

-Consider the graph above ; how many circuits consisting of three edges are there which involve vertex A?

-Consider the graph above ; state the vertex set.

Is it possible to construct a graph consisting of two vertices of even degree and three vertices of odd degree? Explain.

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to put his pencil on the drawing and trace each and every line once and only once, without lifting his pencil. Is John able to complete this task? Explain.

Jim and John decide to play a game. Jim draws a shape consisting only of lines and he asks John to put his pencil on the drawing and trace each and every line once and only once, without lifting his pencil. Is John able to complete this task?

The graph below does not contain an Euler Circuit. If an optimal Eulerization is found for the graph, then how many edges will be traveled over a second time?