Grassmann and Segre varieties over GF(2): some graph theory links

Some recent results (and a conjecture) concerning the polynomial degree of the Grassmannian G1;n;2 of the lines of PG(n; 2) are shown to be equivalent to results (and a conjecture) concerning certain kinds of subgraphs of any (simple) graph = (V; E) which is of order jVj = n+ 1: It turns out that those graphs of size jEj = n = jVj 1 are of particular signi…cance. Somewhat similarly, results concerning the polynomial degree of the Segre variety Sm;n;2 are translated into equivalent assertions concerning certain subgraphs of any graph which is a subgraph of the complete bipartite graph Km+1;n+1: Keywords: polynomial degree, Grassmannian G1;n;2; Segre variety Sm;n;2; subgraph enumerations AMS Classi…cation: 51E20, 05C30, 05C90, 14G25 1 The polynomial degree of a subset of PG(N; 2) In succeeding sections we will be interested in the polynomial degrees of the following varieties over the …nite …eld GF(2): (i) the Grassmann variety G1;n;2 of the lines of PG(n; 2); considered as a subset of points of the …nite projective space PG( n+1 2 1; 2) = P(^Vn+1;2); (ii) the Segre variety Sm;n;2; considered as a subset of points of the …nite projective space PG(mn+m+ n; 2) = P(Vm+1;2 Vn+1;2). However it will help to …rst consider material concerned with the polynomial degree of a general subset of points of a general …nite projective space PG(N; 2) = P(V ); where V = VN+1 = V (N + 1; 2): For the most part the notation will be as in [7]. In particular S = PG(N; 2) denotes the set of points (0-‡ats) of PG(N; 2) = P(V ); and we identify S with the nonzero vectors V n f0g of the vector space V: The set F (V ) of all functions V ! GF(2) is a vector space over GF(2) of dimension