The Sum Number of a The Sum Number of a Disjoint Union of GraphsDisjoint Union of Graphs

Mirka Miller & Joe RyanMirka Miller & Joe Ryan

TheThe University University ofof Newcastle, Australia Newcastle, Australia

W. F. SmythW. F. Smyth

McMaster University, Hamilton, CanadaMcMaster University, Hamilton, Canada

Curtin University, Perth, AustraliaCurtin University, Perth, Australia

Sum LabellingSum Labelling

L : V(G) ℕ.

For u, v V, (u, v) E(G) if and only if

w V such that L(w) = L(u) + L(v).

Sum GraphsSum Graphs

• All sum graphs are disconnected.

• Any graph can be made to support a sum labelling by adding sufficient isolated vertices called isolates.

• The smallest number of isolates required is called the sum number of the graph (σ(G)).

• Sum graphs with this fewest number of isolates are called optimal.

ExampleExampleA Non Optimal LabellingA Non Optimal Labelling

1

3

9

27

81

4

12

36

84

ExampleExampleA Optimal Sum LabellingA Optimal Sum Labelling

1

3

4

7

1114

Potential Perils in Sum LabellingPotential Perils in Sum Labelling

1 2

45

3

6

9

Disjoint Union of GraphsDisjoint Union of Graphs(an example)(an example)

5

9

1317

21

14

38

34

30

26

22

183

4

711

18

21

29

Disjoint Union of GraphsDisjoint Union of Graphs(an example)(an example)

35

63

91119

147

98

266

238

210

182

154

126114

152

266418

684

798

1102

An Upper BoundAn Upper Bound

• σ(G1G2) σ(G1) + σ(G2) – 1

• Inequality is tight for unit graphs

• The technique may be applied repeatedly for a disjoint union of many graphs.

Three Unit Graphs: An ExampleThree Unit Graphs: An Example

14

3

510

7

14

1 2

34

5

1

3 4 5

6

2

Three Unit Graphs: An ExampleThree Unit Graphs: An Example

14

3

510

7

1 2

34

5

14

42 56 70

84

2

Three Unit Graphs: An ExampleThree Unit Graphs: An Example

14

3

510

7

168

252336

420

14

42 56 70

842

A disjoint union of three graphs with sum number 1

A Disjoint Union of A Disjoint Union of pp Graphs Graphs(main result)(main result)

)1(11

pGGp

ii

p

ii

Provided that we can always find a labelin one graph that is co-prime to the largestlabel in one of the others.

Easy if 1 is a label in any of the graphs.

Can we always apply the co-prime Can we always apply the co-prime condition?condition?

• Yes if 1 is a label of any of the graphs.

• No sum graph has yet been found that cannot bear a sum labelling containing 1.

• But…“absence of evidence is not evidence of absence”

Rumsfeld

• Exclusive sum graphs may always be labelled with a labelling scheme containing 1.

Exclusive Sum GraphsExclusive Sum Graphs

• If L is an exclusive sum labelling for a graph G, so is k1L+k2 where k1, k2 are integers such that min(k1L+k2) 1.

Miller, Ryan, Slamin, Sugeng, Tuga (2003)

)1(11

pGGp

ii

p

ii

Provided at least one of the graphs is an exclusive graph

Open QuestionsOpen Questions

1. Can we always find a sum labelling containing the label 1?

2. What is the sum number of a disjoint union of graphs for various families of graphs?

3. What is the exclusive sum number of a disjoint union of graphs for various families of graphs?