Local Orthogonality Dimension

Abstract

An orthogonal representation of a graph G over a field F is an assignment of a vector u v ∈ F t to every vertex v of G, such that u v , u v = 0 for every vertex v and u v , u v ′ = 0 whenever v and v ′ are adjacent in G. The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph G and a given field F, as the smallest possible locality of an orthogonal representation of G over F. This is a local variant of the well-studied orthogonality dimension of graphs, introduced by Lovász (Trans. Inf. Theory, 1978), analogously to the local variant of the chromatic number introduced by Erd ös et al. (Discret. Math., 1986).We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. Such methods are known to imply tight lower bounds on the chromatic number of several graph families of interest, such as Kneser graphs and Schrijver graphs. We prove that graphs for which topological methods imply a lower bound of t on their chromatic number have local orthogonality dimension at least ⌈t/2⌉ + 1 over every field, strengthening a result of Simonyi and Tardos on the local chromatic number (Combinatorica, 2006). We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over R coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal (J. Graph Theory, 2021). As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also study the computational aspects of the local orthogonality dimension and show that for every integer k ≥ 3 and a field F, it is NP-hard to decide whether the local orthogonality dimension of an input graph over F is at most k. We finally present an application of the local orthogonality dimension to the index coding problem from information theory, extending a result of Shanmugam, Dimakis, and Langberg (ISIT, 2013).