2021/10/02 by Inon Attias, Attias, Inon, Ishay Haviv +1
Computer Science · #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #Information Theory (cs.IT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2110.00718
openalex publication_date 2021/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An orthogonal representation of a graph G over a field \mathbbF is an assignment of a vector uv ∈ \mathbbFt to every vertex v of G, such that ⟨ uv,uv ⟩ ≠ 0 for every vertex v and ⟨ uv,uv' ⟩ = 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 \mathbbF, as the smallest possible locality of an orthogonal representation of G over \mathbbF. We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. We prove that graphs for which topological methods imply a lower bound of t on their chromatic number have local orthogonality dimension at least \lceil t/2 \rceil +1 over every field, strengthening a result of Simonyi and Tardos on the local chromatic number. 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. 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 show an NP-hardness result for the local orthogonality dimension and present an application of this graph parameter to the index coding problem from information theory.