The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications
The orthogonality dimension of a graph G=(V,E) over a field ๐ฝ is the smallest integer t for which there exists an assignment of a vector u_v โ๐ฝ^t with โจ u_v,u_v โฉโ 0 to every vertex v โ V, such that โจ u_v, u_v'โฉ = 0 whenever v and v' are adjacent vertices in G. The study of the orthogonality dimension of graphs is motivated by various application in information theory and in theoretical computer science. The contribution of the present work is two-folded. First, we prove that there exists a constant c such that for every sufficiently large integer t, it is ๐ญ๐ฏ-hard to decide whether the orthogonality dimension of an input graph over โ is at most t or at least 3t/2-c. At the heart of the proof lies a geometric result, which might be of independent interest, on a generalization of the orthogonality dimension parameter for the family of Kneser graphs, analogously to a long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976). Second, we study the smallest possible orthogonality dimension over finite fields of the complement of graphs that do not contain certain fixed subgraphs. In particular, we provide an explicit construction of triangle-free n-vertex graphs whose complement has orthogonality dimension over the binary field at most n^1-ฮด for some constant ฮด >0. Our results involve constructions from the family of generalized Kneser graphs and they are motivated by the rigidity approach to circuit lower bounds. We use them to answer a couple of questions raised by Codenotti, Pudlรกk, and Resta (Theor. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field.
READ FULL TEXT