1. Open Problems in Exclusive Sum Labeling Mirka Miller, Joe Ryan, Slamin, Kiki Sugeng, Mauritsius Tuga School of Computer Science and Software Engineering The University of Newcastle- Australia

2. A Graph G(V,E) is a sum graph if there exist a labeling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) + L(v). Every sum graph must contain isolated vertices. If a graph G is not a sum graph, it is always possible to add some finite number of isolated vertices to G to obtain a sum graph. The sum number  (G) of G is the smallest number of isolated vertices that will achieve this result. Sum Graph

3. Sum Graph (Cont’d) The notion of a sum graph was introduced by Harary in 1990 Gould and Rödl proved that there exist graphs G(V,E) such that  (G)   (|V| 2 ). Nagamochi et al proved that this fact should be quite common. More than a decade researchers have failed to produce an example of any such graph.

4. Exclusive Sum Labeling In a sum labeling of a graph G, a vertex w is said to label an edge uv  E(G) if and only if w=u+v. In this case the vertex w is called a working vertex. A sum labeling L is called an exclusive sum labeling with respect to a sub graph H of a graph G if none of the vertices of H are working vertices. We say that H is labeled exclusively The least number r of isolated vertices such that G=H  is a sum graph and H is labeled exclusively is called the exclusive sum number  (H) of graph H.

5. Exclusive Sum Labeling (Cont’d) The notion of an exclusive sum labeling was introduced by Miller et al. Figure 1 and Figure 2 are examples of exclusive sum labeling. Fig.1 Exclusive sum labeling for W 7

6. Exclusive Sum Labeling (Cont’d) Fig.2 Exclusive sum labeling for f 7

7. Facts on Exclusive Sum Labeling Observation 1:  (G)   (G). Observation 2: If L is a sum labeling of a graph G then so is L’ = kL, where k is any positive integer. Observation 3: If L is a sum labeling of a graph H in G=H  then so is the labeling L’ = {k 1 L + k 2 for non working vertices of G, and k 1 L+2k 2 for working vertices of G} where k 1 is any positive integer and k 2 is any integer which results only in positive distinct values in L’. Observation 4: Let  be the maximum degree of a vertex in a graph G. Then  (G)   (G).

8. Exclusive Sum Number for Several Classes of Graphs Graph G  (G)  (G) Stars S n, n  2, 1n Trees, T n, n  3, T n  S n 1? Paths P n 12 Cycles C 4 C n, n  4, 3232 3333 Wheels W 4 W n, n  5, n odd n  6, n even 4 n n/2 +2 4 n Fans F n ?n Friendship graphs f n 22n

9. Exclusive Sum Number for Several Classes of Graphs (Cont’d) Graph G  (G)  (G) Complete graphs K n, n  3 2n-3 Cocktail Party graphs, H 2,n H m,n 4n-5 ? 4n-5 ? Bipartite Graphs, K n,n K m,n  k(n-1)/2 +m/(k-1)  Where k = ????

10. Open Problems Open Problem 1. Find the exclusive sum number for various families of graphs. Open Problem 2. Find the exclusive sum number for disjoint union of graphs. Open Problem 3. Find graphs with exclusive sum number in the order of |V| 2. This is likely to be a very difficult problem. An easier problem might be Open Problem 4. Find graphs with exclusive sum number in the order of |V| t, where t > 1. The caterpillar type of a tree has been shown to have exclusive sum number equal to its maximum degree. These results suggest Conjecture. Every tree has exclusive sum number equal to its maximum degree.

11. Bibliography D. Bergstrand, F. Harary, K. Hodges, G. Jennings, L.Kuklinski and J. Wiener, The sum number of a complete graph, Bull.Malaysian Math. Soc. 12 (1989) 25-28. M. Ellingham, Sum graphs from trees, Ars. Combinatoria 35 (1993) 335-349. R. Gould and V. Rödl, Bounds on the number of isolated vertices in sum graphs, Graph Theory, Combinatorics and Applications,(edited by Y. Alevi, G. Chartrand, O.R. Oellermann and A.J. Schwenk, John Wiley and Sons (1991), 553-562. F. Harary, Sum graphs and difference graphs, Congressus Numerantium 72} (1990) 101-108. N. Harstfield and W.F. Smyth, The sum number of complete bipartite graphs, Graphs and Matrices,(edited by Rolf Rees), Marcel Dekker (1992) 205-211. N. Hartsfiels and W.F. Smyth, A family of sparse graphs of large sum number, Discrete Math. 141 (1995) 163-171. W. He, Y. Shen, L. Wang, Y. Chang, Q. Kang and X. Yu, The integral sum number of complete bipartite graphs K r,s, Discrete Math, 239 (2001), 137-146.

12. Bibliography (Cont’d) M. Miller, J. Ryan, Slamin and W.F. Smyth, Labelling Wheels for minimum sum number, JCMCC, Vol. 28 (1998), 289-297. M. Miller, J. Ryan and W.F. Smyth, The sum labeling for the cocktail party graph, Bulletin of ICA, Vol 22 (1998), 79-90. M. Miller, J. Ryan, Slamin, K. Sugeng and M. Tuga,Exclusive sum graph labeling, preprint, 2003. H. Nagamochi, M.Miller and Slamin, Bounds on the number of isolates in sum graph labeling, Discrete Math. (2001), no 1-3, 219-229. A.V. Pyatkin, New formula for the sum number for the complete bipartite graphs, Discrete Math, 239 (2001), 155-160. A. Sharary, Integral sum graphs from complete graphs, cycles and wheels, Arab Gulf J. Sci. Res. (1996) 1-14. M. Tuga and M. Miller, Optimal exclusive sum labeling for some trees, in preparation. Y. Wang and B. Liu, The sum number and integral sum number of complete bipartite graph, Discrete Math, 239 (2001), 69-82.