Triangles in 2‐factorizations

The triangle‐spectrum for 2‐factorizations of the complete graph Kv is the set of all numbers δ such that there exists a 2‐factorization of Kv in which the total number of triangles equals δ. By applying mainly design‐theoretic methods, we determine the triangle spectrum for all v ≡ 1 or 3 (mod 6), v ≥ 43, as well as for v = 7, 9, 13, 15, 21, and 27. For orders v = 19, 25, 31, 33, 37, 39, we leave only a total of 11 values undecided. To determine the triangle‐spectrum for v ≡ 5 (mod 6) remains an open problem. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 83–94, 1997