Applied Finite Mathematics HW Solutions 82

# Applied Finite Mathematics HW Solutions 82 - 80 EXERCISES...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 80 - EXERCISES 4.3 (d) The edges traced out are shown in the left graph below with arrows indicating direction. It is a path. Since it returns to the same node, it is also a circuit. (e) This is impossible; nodes E and F do not have an edge joining them. ' (f) Since the edge joining nodes G and E is repeated, this is not a path. (g) The edges traced ”out are shown in the right graph below with arrows indicating direction. It is a path. Since it traces every edge of the graph, it is an Euler path. a A s A a c 4‘" G F G 8. (a) Since this graph has four nodes with odd degree, there cannot be an Euler path or circuit. There are many Hamiltonian circuits, one of which is shown in the left ﬁgure below. (b) Since all nodes of the graph have even degree, there are Euler circuits, one of which is shown in the middle ﬁgure below. There is no Euler path. There are many Hamiltonian circuits. (c) Since this graph has exactly two nodes with odd degree, there are Euler paths but no Euler. circuit. One is shown in the right ﬁgure below. There is no Hamiltonian circuit. 9. We use one node for each of the islands and one node for each of the river banks (ﬁgure to the right). Since every node has even degree, there is an Euler circuit for the graph. Hence, it is possible to ' walk around the city starting and ending at the same point and crossing each bridge exactly once. ' ' 10. Suppose we represent rooms and the outside by nodes and doorways by edges (left ﬁgure below). To answer the question, we ask whether there is an Euler path from one node representing the outside to the other node representing the outside. Since all other nodes have even degree, there is indeed an Euler path from outside to outside. ' For the second ﬂoor plan, we again represent rooms and the outside by nodes and doorways . by edges (right ﬁgure above). To answer the question, we ask whether there is an Euler path ' from one node representing the outside to the other node representing the outside. Since there are two nodes with degree three, as well as two nodes with degree one, there is no Euler path from outside to outside. _ _l__2_l L _ _ L--- --_..-.I;.....A :. -..........I.._.... ..:.oL -....-_'..l.c 3-... L..4I..ATY..I..mliu ArltnnkAka DAA‘Idm EnakmmnrinM-im in ..a n. h. l—‘ull L. «spinnumkikﬂazl ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern