Math 455.1 Homework Set 9 Spring, 2009
Due: Monday, May 11
Instructions: Work either individually or in a team.
1. Use the greedy vertex-coloring algorithm to color the vertices of the following graphs.
In each case, tell whether the result is optimal.
a
b
c
d
e
f
HaL
a
b
c
d
e
f
HbL
2. (a) Without actually finding any vertex coloring yet, use the theory we’ve developed
in order to obtain a lower and an upper bound on the chromatic number of the
graph:
1
2
3
4
5
6
(b) Now determine the actual value of χ(G).
3. Finish the solution of our procedure-scheduling problem: Find a minimal vertex col-
oring of the associated graph. Then interpret the results in terms of scheduling the
procedures. (See the handout “Procedure scheduling problem”, 27 April 2009.)
4. Use induction to prove that every tree with exactly nvertices has exactly n−1 edges.
(Note. Be careful: the proposition you want to prove about nis that every tree with
exactly nvertices has exactly n−1 edges.)
5. (a) Find al l spanning trees of the following graph. (The order in which you add
vertices or edges to get a particular spanning tree is irrelevant; all that matters
is what the particular spanning tree is.)
1
2
3
4
5
1
