21-484, Spring 2004, HW 1, Solutions
1. (Problem 1.2.22) Prove that a graph is connected if and only if for
every partition of its vertices into two nonempty sets, there is an edge
with endpoints in both sets.
Necessity. Suppose G is connected. Let S,T be
21-484, Spring 2004, Test 1 Solutions
1. (a) For which m and n is Km,n Eulerian? (State the theorem you are
using and state how you are applying it.)
If G is connected then G is Eulerian if and only if every vertex of
G is of even degree. Since Km,n is co
21-484, Spring 2004, Test 3, Solutions
P oints Score
T otal :
1. Let G be a graph with n = 13 vertices, (G) = 3, (G) = 6, and
= 7. Suppose G is not a complete graph. What are the best upper
and lower bounds that you can put
21-484, Spring 2004, HW 8 Solutions
1. Justify your answers:
(a) (6.1.3) For which r, s is Kr,s planar?
For r 2 or for s 2.
Suppose without loss of generality that r 2. Clearly K1,s is
planar. For a plane drawing connect the center of a circle to s
21-484, Spring 2004, HW 7 Solutions
1. (5.3.4) Prove that the chromatic polynomial of Cn is
(Cn ; k) = (k 1)n + (1)n (k 1).
We prove the statement by induction on n. If n = 3, (Cn ; k) =
(K3 ; k) = k(k 1)(k 2) = k 3 3k 2 + 2k = (k 1)3 + (1)3 (k 1).
21-484, Spring 2004, HW 4 Solutions
1. If m n, an m n array R of numbers is a Latin rectangle if the
entries of R are all from the set cfw_1, 2, . . . , n and if no row contains
a repeated element and no column contains a repeated element. For
21-484, Spring 2004, HW 6 Solutions
1. Starting with the zero ow, produce a maximum value ow in N and
a minimum value s, t cut. Also record each ow-augmenting path you
The algorithm to nd a ow augmenting path produces:
order placed on L :
1 2 3 4 5 6
21-484, Spring 2004, HW 3 Assigned:Jan. 30, Due: Feb 6 (in class)
1. (2.1.7) Prove that every n vertex graph with m edges has at least
m n + 1 cycles.
Suppose G is a connected graph. Choosing a spanning tree T uses n1
edges. Each of the the remaining m n
21-484, Spring 2004, HW 2 (corrected version), Assigned:Jan. 21, Due: Jan.
28 (in class)
1. (1.3.14) Show that every simple graph with at least two vertices has
two vertices of equal degree.
Suppose G is a simple graph with n 2 vertices and degree sequenc
21-484, Spring 2004, Test 2 solutions
1. Let the following complete bipartite graph have the following preferences for their neighbors:
(a) Describe the stable matching algorithm.
If G is a complete bipartite graph with bipartitition X, Y , the