Abstract
In 1987, Alavi, Boals, Chartrand, Erdös, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). Though several classes of graphs have been shown to have an ASD, the conjecture remains open. In this paper, we investigate the similar problem for tournaments. In particular, using Kirkman Triple Systems, we will show that all tournaments of order 6n + 3 have an ASD.
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Alavi Y., Boals A.J., Chartrand G., Erdös P., Oellermann O.: The ascending subgraph decomposition problem. Congr. Numer. 58, 7–14 (1987)
Chartrand G., Lesniak L.: Graphs & Digraphs, 4th ed. Chapman & Hall/CRC, Boca Raton (2005)
Ray-Chaudhuri, D.K., Wilson, R.M.: Solution to Kirkman’s schoolgirl problem. Combinatorics 19, 187–203 (1971) (Proceedings of Symposia in Pure Mathematics, University of California, Los Angeles, CA, 1968)
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Wagner, B.C. Ascending Subgraph Decompositions of Tournaments of Order 6n + 3. Graphs and Combinatorics 29, 1951–1959 (2013). https://doi.org/10.1007/s00373-012-1236-1
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DOI: https://doi.org/10.1007/s00373-012-1236-1