In this paper, the chromatic polynomial structure on Riemannian manifolds and the almost golden structure on the tangent bundle of a Finsler manifold have been studied. A class of g-natural metrics on the tangent bundle of a Finsler manifold have been considered and some conditions under which the golden structure is compatible with the above-mentioned metric are proposed. The Levi-Civita connection associated with the mentioned metric is calculated and the results of it are presented. Finally, the integrability of the golden structure and its compatibility with the covariant derivative is studied.