The number of triangles in 2-factorizations of K2n minus a 1-factor

Abstract

A 2-factorization of K_n is a partition of the edge set of K _n into 2-factors. Given an arbitrary 2-factorization F = {F ₁,F₂,... F_n-1} of K_2n,let δ_i,be the number of triangles contained in F_i and let δ = Σδ_i. Then F is said to be a 2-factorization with δ triangles. Denote by Δ(2ra),the set of all δ such that there exists a 2-factorization with 6 triangles. Let (equation) where (equation). In this paper,we consider the problem of constructing 2-factorization of K_2n containing a specified number of triangles. We proved that apart from some exceptions Δ(2n) = PΔ(2n).