... the first coloring, receive distinct colors inthe second coloring. For a survey of the results on orthogonal edge colorings the readeris referred to [1]. The results on orthogonal edge coloring ... <n,wemusthavex>√n .4 Strong orthogonal coloringsA k-strong orthogonal coloring, is a k-orthogonal coloring c1, ,ck,whereforeachvertexv and for every pair of distinct colorings ciand cj, ci(v) ... proper coloring of Gr,rsince Lis a Latin square. Also, c2is a proper coloring of Gr,rsince Ltis a Latin square and since{(1, 1), ,(r, r)} are an independent set in Gr,r. Also, the colorings...