MATH 310, FALL 2003
(Combinatorial Problem
Solving)
Lecture 5,Wednesday, September 10
A.2 Mathematical Induction


Let pn denote a statement involving n
objects.
Induction proof of pn, for all n ¸ 0:
• Initial step (Induction Basis): Verify that p0 is true.
• Induction step: Show that if p0, p1, ..., pn-1 are true,
then pn must be true.


Note: You have to prove p0. You also have
to prove pn, but in the proof you may
“pretend” that pn-1 or any other pk, k < n is
true.
Note: Induction comes in various forms. For
instance, sometimes the initial step involves
some other small number, say, p1, or p3, ...
Example 1




Let sn = 1 + 2 + ... + n. (A)
Prove sn = n(n+1)/2. (B)
Proof by induction.
Initial step:



s1 = 1. (A).
s1 = 1(1+1)/2 = 1 (B).
Induction step:



sn = [1 + 2 + ... + (n-1)] + n = sn-1 + n.
Now assume sn-1 = (n-1)n/2
sn = sn-1 + n = (n-1)n/2 + n = n(n+1)/2.
2.1 Euler Cylces

Homework (MATH 310#2W):
• Read 2.2. Read Supplement I.(pp 46-48) Write down a
list of all newly introduced terms (printed in boldface or
italic)
• Do Exercises A.2: 4,12,17,24
• Do Exercises 2.1: 2,10,12,17
• Volunteers:
• ____________
• ____________
• Problem: 2.1:17.
Challenge (up to 5 + 5 points): Do
Exerecise 2.1: # 20 (requires computer
programming).
Multigraph

A


B
C
D
In a multigraph we
may have:
Parallel edges
Loop edges (=
loops).
Königsberg Bridges

A
B
C
D
Great Swiss
mathematician
Leonhard Euler
solved the problem
of Seven Bridges
of Königsberg by
showing that it is
impossible to walk
across each bridge
just once.
Trails and Cycles




Path P = x1- x2 - ... – xn (all vertices
distinct).
Circuit C = x1- x2 - ... – xn – x1 [a path
with an extra edge (xn ,, x1 )].
Trail T = x1- x2 - ... – xn (vertices may
repeat but all edges are distinct).
Cycle E = x1- x2 - ... – xn – x1 [a trail with
an extra edge (xn ,, x1 )].
Euler Cycles and Trails
A cycle that uses every edge of a
graph is called an Euler cycle (and
visits every vertex).
 A trail that uses every edge of a graph
is called an Euler trail (and visits every
vertex at least once).

Theorem 1 (Euler, 1736)

An (undirected) multigraph has an
Euler cycle if and only if:
• it is connected and
• has all vertices of even degree.
Example 3: Routing Street
Sweepers

Solid red edges
represent a
collection of blocks
to be swept.
Corollary

A multigraph has an Euler trail, but
not an Euler cycle, if and only if
it is connected and
 has exactly two vertices of odd degree.
