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

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Presentation transcript:

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

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).

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.

Example A Non Optimal Labelling

Example A Optimal Sum Labelling

Potential Perils in Sum Labelling

Disjoint Union of Graphs (an example)

Disjoint Union of Graphs (an example)

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.

Three Unit Graphs: An Example

Three Unit Graphs: An Example

Three Unit Graphs: An Example A disjoint union of three graphs with sum number 1

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.

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.

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

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?