Edge Grundy numbers of the regular Turán graphs

Suppose r ≥ 3 and k ≥ 2 are integers. If k(r − 1) is even then the edge Grundy number of the Turán graph Tkr,r is 2k(r − 1) − 1. Otherwise, the edge Grundy number of Tkr,r is 2k(r − 1) − m for some m ∈ {1, . . . ,min[g(k), 2r − 1]}, where g(k) = 2 if k = 3 and g(k) = k − 2 otherwise.