Rigidity of Balanced Minimal Cycle Complexes
Ryoshun Oba · SIAM Journal on Discrete Mathematics · 2025
Abstract. A [Formula: see text]-dimensional simplicial complex [Formula: see text] is balanced if its graph [Formula: see text] is [Formula: see text]-colorable. Klee and Novik [ Mathematika, 62 (2016), pp. 441–477] established the balanced lower bound theorem for balanced normal [Formula: see text]-pseudomanifolds [Formula: see text] with [Formula: see text] by showing that for any set of three colors [Formula: see text], the subgraph of [Formula: see text] induced by the vertices of color in [Formula: see text] is rigid in [Formula: see text]. We prove that the same rigidity result—and consequently the balanced lower bound theorem—holds for balanced minimal [Formula: see text]-cycle complexes with [Formula: see text]. Motivated by the work of Stanley [ Trans. Amer. Math. Soc., 249 (1979), pp. 139–157; Graphs Combin., 3 (1987), pp. 55–66] on colored systems of parameters for the Stanley–Reisner ring of balanced simplicial complexes, we further investigate the infinitesimal rigidity of nongeneric realizations of balanced and, more generally, [Formula: see text]-balanced simplicial complexes. Among other results, we show that for [Formula: see text], a balanced connected homology [Formula: see text]-manifold can be realized as an infinitesimally rigid framework in [Formula: see text] such that each vertex of color [Formula: see text] lies on the [Formula: see text]th coordinate axis.