Graph Theory Proof Techniques and Properties, Slides of Computer Science

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Proof Techniques

Proof Techniques

Induction

Induction

If If

u u

and and

v v

are distinct vertices in G, then are distinct vertices in G, then

every every

u,v u,v-

-walk

walk

in G contains a in G contains a

u,v u,v-

-path.

path.

Every closed odd walk contains an odd Every closed odd walk contains an odd

cycle. cycle.

Contradiction

Contradiction

We prove A We prove A

B by showing that B by showing that “

“A true and

A true and

B false B false”

” is impossible

is impossible

Suppose G has a vertex set { Suppose G has a vertex set {

v v

1 1

, …, v

v

n n

}, with

}, with

n n

3. If at least two of the
4. If at least two of the subgraphs

subgraphs from

from

G

G

v

v

1 1

…, G

, G

v v

n n

are connected, then G is are connected, then G is

connected connected

A graph is bipartite A graph is bipartite iff

iff it has no odd cycle

it has no odd cycle

Extremality

Extremality

If G is a simple graph in which every vertex If G is a simple graph in which every vertex

degree is at least degree is at least

k k

, then G contains a path , then G contains a path

of length at least of length at least

k. k.

• If

If

k k

≥ ≥

2 2

, then G also contains a cycle of length at , then G also contains a cycle of length at

least least

k+ k+

. .

If G is a nontrivial graph and has no cycle, If G is a nontrivial graph and has no cycle,

then G has a vertex of degree 1. then G has a vertex of degree 1.

Every nontrivial graph has at least two Every nontrivial graph has at least two

vertices that are not cut vertices. vertices that are not cut vertices.