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 EnakmmnrinMim in ..a n. h. l—‘ull L. «spinnumkikﬂazl ...
View
Full Document
 Fall '11
 Dr.Cornell
 Math

Click to edit the document details