A Note on Varieties of Groupoids Arising from m-Cycle Systems
Darryn E. Bryant
DOI: 10.1023/A:1022423910787
Abstract
Decompositions of the complete graph with n vertices K n into edge disjoint cycles of length m whose union is K n are commonly called m-cycle systems. Any m-cycle system gives rise to a groupoid defined on the vertex set of K n via a well known construction. Here, it is shown that the groupoids arising from all m-cycle systems are precisely the finite members of a variety (of groupoids) for m = 3 and 5 only.
Pages: 197–200
Keywords: $m$-cycle system; variety; equationally denned; groupoid
Full Text: PDF
References
1. D.E. Bryant and C.C. Lindner, "2-perfect m-cycle systems can be equationally defined for m - 3, 5, and 7 only," Algebra Universalis (to appear).
2. S. Burris and H.P. Sankappanavar, "A course in universal algebra," Graduate Texts in Mathematics 78, Springer-Verlag, New York, 1981.
3. F.K. Hwang and S. Lin, "Neighbour designs," J. Combin. Theory Ser. A 23 (1977), 302-313.
4. A. Kotzig, "Groupoids and partitions of complete graphs," Combinatorial Structures and Their Applications (Proc. Calgary Internal. Conf., Calgary, Alta), Gordon and Breach, New York (1970), 215-221.
5. C.C. Lindner and C.A. Rodger, "Decompositions into cycles II: Cycle systems," in Contemporary design theory: a collection of surveys (eds. J.H. Dinitz and D.R. Stinson), John Wiley and Sons, New York (1992), 325-369.
6. C.C. Lindner, C.A. Rodger, and D.R. Stinson, "Embedding cycle systems of even length," J. Combin. Math. Combin. Comput. 3 (1988), 65-69.
7. R.M. Wilson, "Decompositions of edge-colored complete graphs," Journal of Combinatorial Designs (to appear).
2. S. Burris and H.P. Sankappanavar, "A course in universal algebra," Graduate Texts in Mathematics 78, Springer-Verlag, New York, 1981.
3. F.K. Hwang and S. Lin, "Neighbour designs," J. Combin. Theory Ser. A 23 (1977), 302-313.
4. A. Kotzig, "Groupoids and partitions of complete graphs," Combinatorial Structures and Their Applications (Proc. Calgary Internal. Conf., Calgary, Alta), Gordon and Breach, New York (1970), 215-221.
5. C.C. Lindner and C.A. Rodger, "Decompositions into cycles II: Cycle systems," in Contemporary design theory: a collection of surveys (eds. J.H. Dinitz and D.R. Stinson), John Wiley and Sons, New York (1992), 325-369.
6. C.C. Lindner, C.A. Rodger, and D.R. Stinson, "Embedding cycle systems of even length," J. Combin. Math. Combin. Comput. 3 (1988), 65-69.
7. R.M. Wilson, "Decompositions of edge-colored complete graphs," Journal of Combinatorial Designs (to appear).