The Hamilton–Waterloo Problem for Triangle-Factors and Heptagon-Factors

Given 2-factors (Formula presented.) and $$S$$S of order $$n$$n, let $$r$$r and $$s$$s be nonnegative integers with (Formula presented.), the Hamilton–Waterloo problem asks for a 2-factorization of (Formula presented.) if (Formula presented.) is odd, or of (Formula presented.) is even, in which (Formula presented.) of its 2-factors are isomorphic to (Formula presented.) and the other (Formula presented.) 2-factors are isomorphic to (Formula presented.). In this paper, we solve the problem for the case of triangle-factors and heptagon-factors for odd $$n$$n with 3 possible exceptions when (Formula presented.).