A combinatorial approach to the orthogonality on critical orbital sets

✍ Scribed by J.A. Dias da Silva; Maria M. Torres

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 199 KB
• Volume: 414
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2005.10.030

✦ Synopsis

Let G = (X, Y, E) be a bipartite multigraph. Let µ = (µ 1 , . . . , µ s ) be a partition of |E|. A µ-coloring for G is a proper edge coloring (U 1 , . . . , U s ), such that |U i | = µ i , i = 1, . . . , s. Let ρ X be the partition of |E| whose terms are the degrees of the vertices of X arranged in non-increasing order and let ρ X be its conjugate partition. A necessary condition for the existence of a ρ X -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.