David Jenkins Advisor: Dr. DeLaViña

Domination number (γ)

2

Total Domination number (γt)

Three Domination number (γ3)

Two Domination number (γ2)

Cockayne, E.J., Dawes, R.M., Hedetniemi,

S.T., Total domination in graphs. Networks,

10:211-219, 1980.

3

Computer program written by Dr. Ermelinda

DeLaViña

Creates conjectures for specific graph

parameters

Total domination for at most cubic connected graphs

4

Graph theory directed study

Introduced to terms, notation, and proof

techniques

First conjectures from Graffiti.pc

)(Gt

t The number of minimum degree vertices

5

6

2t 2t

)(Gt )(# voft

1 G 1)(# vof

Cockayne, E.J., Dawes, R.M., & Hedetniemi,

S.T., Total domination in graphs. Networks

10:211-219, 1980.

Total domination not bounded by 2

domination

Paths and cycles are some exceptions

Interestingly Graffiti.pc conjectured the

following:

7

nt3

2

Conjecture. Let G be a connected at most

cubic graph. Then

Simplified as

Several partial results have been found

2

)(*2)( 3 G

Gt

3 t

8

If , then

If G is at most cubic then,

33

2n

9

32

n

)()( 3 GGt

Conjecture (Jenkins) Let G be an at most

cubic connected n-vertex graph.

Then G is 3-regular if and only if

The implication

if G is 3-regular, then

Is false as seen in the graph above.

)(2

3 Gn

10

)(2

3 Gn

Proposition. Let G be an at most cubic

connected n-vertex graph.

If , then G is 3-regular

Follow-up question

For a graph that is not restricted

to at most cubic, the above graph

is simple counterexample

2)(3

nG

11

Proposition. Let be a smallest 3-

dominating set for G. If there is only one

not dominated by a vertex in

then

12

3D

3Dv3D

3 t

13

3-dominating set

Total dominating set

14

3-dominating set

Error set

Disproved 25 different conjectures from

Graffiti.pc

Proved several partial results for

Found a partial result in a 2010 paper of Chellali

et. Al, If G is an at most cubic tree different

than a star, then

Future work could improve ‘swapping’ technique

15

3 t

23 t

16