IWOCA 2010: Combinatorial Algorithms pp 238-241

# On a Relationship between Completely Separating Systems and Antimagic Labeling of Regular Graphs

• Oudone Phanalasy
• Mirka Miller
• Leanne Rylands
• Paulette Lieby
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

## Abstract

A completely separating system (CSS) on a finite set [n] is a collection $$\mathcal C$$ of subsets of [n] in which for each pair a ≠ b ∈ [n], there exist $$A, B\in\mathcal C$$ such that a ∈ A, b ∉ A and b ∈ B, a ∉ B.

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2, ..., q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling.

In this paper we show that there is a relationship between CSSs on a finite set and antimagic labeling of graphs. Using this relationship we prove the antimagicness of various families of regular graphs.

## Keywords

completely separating system vertex antimagic edge labeling antimagic labeling regular graph

## References

1. 1.
Alon, N., Kaplan, G., Lev, A., Roditty, Y., Yuster, R.: Dense graphs are antimagic. J. Graph Theory 47(4), 297–309 (2004), http://dx.doi.org/10.1002/jgt.20027
2. 2.
Bača, M., Miller, M.: Super Edge-Antimagic Graphs: a Wealth of Problems and Some Solutions. BrownWalker Press, Boca Raton (2008)Google Scholar
3. 3.
Bloom, G.S., Golomb, S.W.: Applications of numbered undirected graphs. Proc. IEEE 65, 562–570 (1977)
4. 4.
Bloom, G.S., Golomb, S.W.: Numbered complete graphs, unusual rulers, and assorted applications. In: Theory and Applications of Graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976). Lecture Notes in Math., vol. 642, pp. 53–65. Springer, Berlin (1978)Google Scholar
5. 5.
Cheng, Y.: A new class of antimagic Cartesian product graphs. Discrete Math. 308(24), 6441–6448 (2008), http://dx.doi.org/10.1016/j.disc.2007.12.032
6. 6.
Cranston, D.W.: Regular bipartite graphs are antimagic. J. Graph Theory 60(3), 173–182 (2009), http://dx.doi.org/10.1002/jgt.20347
7. 7.
Dickson, T.J.: On a problem concerning separating systems of a finite set. J. Combinatorial Theory 7, 191–196 (1969)
8. 8.
Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin. 16($$\sharp$$DS6) (2009)Google Scholar
9. 9.
Hartsfield, N., Ringel, G.: Pearls in graph theory: a comprehensive introduction. Academic Press Inc., Boston (1990)
10. 10.
Phanalasy, O., Roberts, I., Rylands, L.: Covering separating systems and an application to search theory. Australas. J. Combin. 45, 3–14 (2009)
11. 11.
Ramsay, C., Roberts, I.T.: Minimal completely separating systems of sets. Australas. J. Combin. 13, 129–150 (1996)
12. 12.
Ramsay, C., Roberts, I.T., Ruskey, F.: Completely separating systems of k-sets. Discrete Math. 183(1-3), 265–275 (1998)
13. 13.
Roberts, I.T.: Extremal Problems and Designs on Finite Sets. Ph.D. thesis, Curtin University of Technology (1999)Google Scholar
14. 14.
Roberts, I., D’Arcy, S., Gilbert, K., Rylands, L., Phanalasy, O.: Separating systems, Sperner systems, search theory. In: Ryan, J., Manyem, P., Sugeng, K., Miller, M. (eds.) Proceedings of the Sixteenth Australasian Workshop on Combinatorial Algorithms, pp. 279–288 (September 2005)Google Scholar
15. 15.
Wang, T.M.: Toroidal grids are anti-magic. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 671–679. Springer, Heidelberg (2005)
16. 16.
Wang, T.M., Hsiao, C.C.: On anti-magic labeling for graph products. Discrete Math. 308(16), 3624–3633 (2008)
17. 17.
Zhang, Y., Sun, X.: The antimagicness of the cartesian product of graphs. Theor. Comput. Sci. 410(8-10), 727–735 (2009)

## Authors and Affiliations

• Oudone Phanalasy
• 1
• 2
• Mirka Miller
• 1
• 3
• 4
• 5
• Leanne Rylands
• 6
• Paulette Lieby
• 7
1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
2. 2.Department of MathematicsNational University of LaosVientianeLaos
3. 3.Department of MathematicsUniversity of West BohemiaPilsenCzech Republic
4. 4.Department of Computer ScienceKing’s College LondonUK
5. 5.Department of MathematicsITB BandungIndonesia
6. 6.School of Computing and MathematicsUniversity of Western SydneyAustralia
7. 7.NICTACanberraAustralia