Summary
The existence of edge-coloured block designs with block size four is studied for all nonisomorphic colourings of the edges of aK 4. There are 25 nonisomorphic edge-colouredK 4's; for each, we examine the existence of edge-coloured designs with the minimum possible index. Uniform cases lead to block designs, perpendicular arrays, nested Steiner triple systems, idempotent Schroeder quasigroups, and other combinatorial objects.
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References
Bermond, J.-C. andSotteau, D.,Graph decompositions and G-designs. In:Proc. Fifth British Combinatorial Conf. (Univ. Aberdeen, 1975). Utilitas Math., Winnipeg, 1976, pp. 53–72.
Beth, T., Jungnickel, D. andLenz, H.,Design theory. Bibliographisches Institut, Mannheim, 1985.
Brayton, R. K., Coppersmith, D. andHoffman, A. J.,Self-orthogonal Latin squares of all orders n ≠ 2, 3,or 6. Bull. Amer. Math. Soc.80 (1974), 116–118.
Brouwer, A. E., Hanani, H. andSchrijver, A.,Group divisible designs with block size four. Discrete Math.20 (1977), 1–10.
Gibbons, P. B.,Computing techniques for the construction and analysis of block designs. Ph.D. thesis, University of Toronto, 1975.
Granville, A., Mosiadis, A. andRees, R.,On complementary decompositions of the complete graph (to appear).
Hanani, H.,The existence and construction of balanced incomplete block designs. Ann. Math. Statist.32 (1961), 361–386.
Hanani, H. Ray Chaudhuri, D. K. andWilson, R. M.,On resolvable designs. Discrete Math.3 (1972), 343–357.
Hell, P. andRosa, A.,Graph decompositions, handcuffed prisoners and balanced p-designs. Discrete Math.2 (1972), 229–252.
Kirkman, T. P.,On a problem in combinations. Cambridge and Dublin Math. M.2 (1847), 191–204.
Lindner C. C., Mendelsohn, N. S. andSun, S. R.,On the construction of Schroeder quasigroups. Discrete Math.32 (1980), 271–280.
Lindner C. C., Mullin, R. C. andStinson D. R.,On the spectrum of resolvable orthogonal arrays invariant under the Klein group K 4. Aequationes Math.26 (1983), 176–183.
Lindner, C. C., Mullin, R. C. andvan Rees G. H. J.,Separable orthogonal arrays. Utilitas Math.31 (1987), 24–32.
Mullin, R. C., Schellenberg, P. J., van Rees, G. H. J. andVanstone S. A.,On the construction of perpendicular arrays. Utilitas Math.18 (1980), 141–160.
Stinson, D. R.,The spectrum of nested Steiner triple systems. Graphs Combins.1 (1985), 189–191.
Wilson, R. M.,An existence theory for coloured block designs. Presented at the Colloquium on Algebraic Combinatorics, Montreal, 1986.
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Colbourn, C.J., Stinson, D.R. Edge-coloured designs with block size four. Aeq. Math. 36, 230–245 (1988). https://doi.org/10.1007/BF01836093
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DOI: https://doi.org/10.1007/BF01836093