1. The Sum Number of a Disjoint Union of Graphs Mirka Miller & Joe Ryan The University of Newcastle, Australia W. F. Smyth McMaster University, Hamilton, Canada Curtin University, Perth, Australia

2. Sum 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).

3. Sum 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.

4. Example A Non Optimal Labelling 1 3 9 27 81 4 12 36 84

5. Example A Optimal Sum Labelling 1 3 4 7 11 14

6. Potential Perils in Sum Labelling 1 2 4 5 3 6 9

7. Disjoint Union of Graphs (an example) 5 9 13 17 21 14 38 34 30 26 22 18 3 4 7 11 18 21 29

8. Disjoint Union of Graphs (an example) 35 63 91 119 147 98 266 238 210 182 154 126 114 152 266 418 684 798 1102

9. An Upper Bound σ(G 1  G 2 )  σ(G 1 ) + σ(G 2 ) – 1 Inequality is tight for unit graphs The technique may be applied repeatedly for a disjoint union of many graphs.

10. Three Unit Graphs: An Example 1 4 3 5 10 7 14 1 2 34 5 1 3 4 5 6 2

11. Three Unit Graphs: An Example 1 4 3 5 10 7 1 2 34 5 14 42 56 70 84 2

12. Three Unit Graphs: An Example 1 4 3 5 10 7 168 252336 420 14 42 56 70 84 2 A disjoint union of three graphs with sum number 1

13. A Disjoint Union of p Graphs (main result) Provided that we can always find a label in one graph that is co-prime to the largest label in one of the others. Easy if 1 is a label in any of the graphs.

14. Can we always apply the co-prime 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.

15. Exclusive Sum Graphs If L is an exclusive sum labelling for a graph G, so is k 1 L+k 2 where k 1, k 2 are integers such that min(k 1 L+k 2 )  1. Miller, Ryan, Slamin, Sugeng, Tuga (2003) Provided at least one of the graphs is an exclusive graph

16. Open 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?