An orthogonal representation of a graph $G$ over a field $\mathbb{F}$ is an assignment of a vector $u_v \in \mathbb{F}^t$ to every vertex $v$ of $G$, such that $\langle u_v,u_v \rangle \neq 0$ for every vertex $v$ and $\langle u_v,u_{v'} \rangle = 0$ 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 $\mathbb{F}$, as the smallest possible locality of an orthogonal representation of $G$ over $\mathbb{F}$. 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 $\mathbb{R}$ 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 $\mathsf{NP}$-hardness result for the local orthogonality dimension and present an application of this graph parameter to the index coding problem from information theory.

翻译：图 $G$ 在域 $\mathbb{F}$ 上的正交表示是将向量 $u_v \in \mathbb{F}^t$ 分配给 $G$ 的每个顶点 $v$，使得对于每个顶点 $v$ 有 $\langle u_v,u_v \rangle \neq 0$，并且当 $v$ 与 $v'$ 在 $G$ 中相邻时，有 $\langle u_v,u_{v'} \rangle = 0$。正交表示的局部性是图中闭邻域关联向量所生成子空间的最大维数。我们引入了一个新的图参数，称为局部正交性维数，对于给定图 $G$ 和域 $\mathbb{F}$，定义为 $G$ 在 $\mathbb{F}$ 上正交表示的最小可能局部性。我们研究了拓扑方法在证明局部正交性维数下界方面的有效性。我们证明，对于拓扑方法能推断其色数下界为 $t$ 的图，其局部正交性维数在每个域上至少为 $\lceil t/2 \rceil +1$，这加强了Simonyi和Tardos关于局部色数的结果。我们证明对于某些图这个下界是紧的，而对于其他图，实数域上的局部正交性维数等于色数。更一般地，我们证明对于每个线图的补图，实数域上的局部正交性维数与色数一致。这加强了Daneshpajouh、Meunier和Mizrahi最近的结果，他们证明了这些图的局部色数和标准色数相等。作为他们结果的另一个推广，我们证明来自Kneser图族的一些额外图的局部色数和标准色数也相等。我们还展示了局部正交性维数的$\mathsf{NP}$-困难性结果，并将此图参数应用于信息论中的索引编码问题。