기초과학연구원 이산수학그룹
On November 21, 2019, Frédéric Meunier from École Nationale des Ponts et Chaussées, Paris presented a talk on the orthogonal representations and the orthogonality dimension of graphs at the discrete math seminar. The title of his talk was “Topological bounds for graph representations over any field“.
Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb {R}$. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb {R}$ – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.
This is joint work with Meysam Alishahi.