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AMS/MBA 546 (Fall 2010)
Estie Arkin
Network Flows
Homework Set # 4
Due Thursday, October 7, 2010.
1). Let
G
= (
V, E
) be a connected multigraph in which every vertex has degree 4. Show that
G
can be
decomposed into two graphs, each containing all nodes, such that the degree of every vertex is 2 in each
graph. In other words, show that there exist
E
1
and
E
2
, where
E
i
⊂
E
,
E
1
∪
E
2
=
E
,
E
1
∩
E
2
=
∅
and
G
1
= (
V, E
1
),
G
2
= (
V, E
2
) have all nodes of degree 2.
2). (a). Prove that if a directed graph has an Euler cycle then the directed graph is strongly connected.
(b). Is the converse true? (Does every strongly connected directed graph have an Euler cycle?) Prove or
give a counterexample.
3). The
k
th power
G
k
of a graph
G
is a graph with the same set of vertices as
G
, two vertices have an edge
between them in
G
k
if and only if they are joined by a path of at most
k
edges in
G
. We claimed in class
that if
G
is biconnected then
G
2
has a Hamilton cycle. (This is non trivial to show!)
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Arkin,E

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