Periodic gossiping in back-to-back trees

The d-ary back-to-back tree of height k, BBTdk, consists Of two complete d-ary trees of height k the leaves of which are identified. For a proper coloring of the edges of BBTdk with c greater than or equal to d + 1 colors, we consider periodic gossiping, i.e. full-duplex all-to-all broadcasting in the 1-port model where communication is made on a link of color i at any time (round) = i mod c. We present a coloring for which this process finishes within k periods, where a period is the collection of c consecutive rounds. We prove that this is optimal for c = d + 1.