On Minrank and Forbidden Subgraphs

Ishay Haviv, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
The minrank over a field F of a graph G on the vertex set {1, 2, . . . , n} is the minimum possible rank of a matrix M ∈ F n×n such that M i,i = 0 for every i, and M i,j = 0 for every distinct nonadjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n, H, F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this paper we study this quantity for various graphs H and fields F. For finite fields, we prove by a
more » ... abilistic argument a general lower bound on g(n, H, F), which yields a nearly tight bound of Ω( √ n/ log n) for the triangle H = K 3 . For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n, H, R) ≥ n δ for some δ = δ(H) > 0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudlák, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.
doi:10.4230/lipics.approx-random.2018.42 dblp:conf/approx/Haviv18a fatcat:d732lo7pjfdqjofh7iu2aijlri