List decomposition of graphs

A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is the same. In particular, any family of trees has the common gcd property. Let F = {H₁,...,H_r} be a family of graphs having the common gcd property, and let d be the common gcd. It is proved that there exists a constant N = N(F) such that for every n > N for which d divides n -1, and for every equality of the form α₁e(H₁) +⋯+α_re(H_r)= (n/2), where α₁,...,α_r are nonnegative integers, the complete graph K_n has a decomposition in which each H_i appears exactly α_i times. In case F is a family of trees the bound N(F) is shown to be polynomial in the size of F, and, furthermore, a polynomial (in n) time algorithm which generates the required decomposition is presented.