The Triangle Spectrum For Complete Graph K_2n

Suppose F is a 1-factor of K2n. Given an arbitrary 2-factorization F={F1, F2,…, Fn-1} of K2n\F, letδi be the number of triangles contained in Fi and letδ=∑δi. Then F is said to be a 2-factorization with 6 triangles. The triangle spectrum for complete graph K2n is the set of allδsuch that there exists a 2-factorization withδtriangles, denoted byΔ(2n). Let whereIn this paper, we consider the problem of establishing the triangle spectrum for complete graph K2n. We proved that apart from some small exceptions and some possible exceptions,Δ(2n)= PΔ(2n).