This thesis focuses on disjunctive constraints and related combinatorial optimization problems: minimum biclique partition problem and minimum biclique cover problem. We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. An NP-complete combinatorial optimization problem, the minimum biclique cover problem, needs to be solved in the combinatorial procedure. It motivates us to study minimum biclique partition and biclique cover problems on co-chordal graphs, which is the complementary graph of chordal. We provide heuristics to find biclique partitions on co-chordal graphs and show that the heuristics can provide biclique partitions close to optimal or even optimal if the graphs satisfy further assumptions on the structures. In addition, we provide a tighter bound for minimum biclique covers on a subclass of co-chordal graphs and present a lower bound for minimum biclique covers of general graphs. We also investigate the disjunctive constraints for piecewise linear relaxations of univariate and high-dimensional nonlinear functions that could appear in optimization problems. In order to study those piecewise linear relaxations, we introduce a new class of combinatorial disjunctive constraints: generalized n-dimensional ordered CDCs and present logarithmically sized ideal formulations under the independent branching scheme.