A combinatorial approach to the orthogonality on critical orbital sets

Let G = (X, Y, E) be a bipartite multigraph. Let μ = (μ1, … , μs) be a partition of ∣E∣. A μ-coloring for G is a proper edge coloring (U1, … , Us), such that ∣Ui∣ = μi, i = 1, … , s. Let ρX be the partition of ∣E∣ whose terms are the degrees of the vertices of X arranged in non-increasing order and let be its conjugate partition. A necessary condition for the existence of a -coloring for G is proved.An application of this necessary condition to the study of the orthogonality of critical symmetrized decomposable tensors is presented. As a consequence, a lower bound for the orthogonal dimension of any critical orbital set is computed.Finally, a conjecture about the non-orthogonality of a class of critical symmetrized decomposable tensors associated with square partitions, which is equivalent to a conjecture of Huang and Rota on Latin squares, is established.