Optimal boxicity of transitive closures of hypercubes and divisor graphs

Determine the optimal value of the boxicity function f(d)=box(TC(H_d)) for the transitive closure TC(H_d) of the d-dimensional hypercube, and determine the exact boxicity of the divisor graph D(n) for every natural number n.

Background

The paper identifies the divisor graph D(n), for n with prime factorization n=p_1^{a_1}⋯p_s^{a_s}, with the transitive closure TCC(a_1,…,a_s) of a Cartesian product of complete graphs. In the squarefree case, D(n) is isomorphic to TC(H_d), the transitive closure of the d-dimensional hypercube. The authors define f(d)=box(TC(H_d)) and prove the lower bound f(d)≥d/2.

The general lower bound for boxicity of TCC(m_1,…,m_d) depends on the values of f(d), f(d−1),…,f(2). Consequently, improving the lower bound or determining f(d) exactly would sharpen the paper’s results for divisor graphs. The authors explicitly also ask for the exact boxicity of D(n).