Math 455.1 Homework Set 8 Spring, 2009
Due: Monday, May 4
Instructions: Work either individually or in a team.
1. Use our algorithmic method of proof (from class) of Euler’s Theorem to construct
an Eulerian trail from vertex ato ain the following graph:
a
f
e
d
c
b
2. (a) Use the function EulerianQ from to determine for which n= 2,3,...,10 the cube
graph Qnis Eulerian. Nicely format your results as a Mathematica table without
directly entering the individual entries of the table. Note: In Combinatorica
notation, the graph Qnis Hypercube[n].
(b) In general, for which values of nis the graph Qnis Eulerian? Formulate a
conjecture based upon your results from (a).
(c) Now prove that your conjecture is true in general.
3. (a) For each of the graphs Aand Bshown below, do the hypotheses of Ore’s Theorem
about Hamiltonian graphs hold?
a
b
c
d
e
A
a
b
c
d
e
B
(b) Determine whether each of the graphs in (a) is Hamiltonian and indicate why or
why not.
4. In class you saw a method that constructs, from the reflective Gray code of n-bit binary
words, the reflective Gray code of (n+ 1)-bit binary words. Apply that method to
construct the reflective Gray code of 5-bit binary words starting with the following
reflective Gray code of 4-bit binary words, shown here (horizontally, to save space).
0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000
1
