The edges of highly-connected symmetrical graphs are colored so that they form Hamiltonian cycles. As an introduction we discuss the coloring of the complete graphs K2m+ 1 for m> 1, but the focus is on the graphs resulting from symmetrical perspective projections of the edges of the regular 4-dimensional polytopes into 3-space. The goal is to color all edges in these graphs with multiple congruent copies of Hamiltonian cycles exhibiting as much symmetry as possible.