On the ideal of orthogonal representations of a graph in R2

Abstract

In this paper, we study orthogonal representations of simple graphs G in R d from an algebraic perspective in case d = 2. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to R d where nonadjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for d = 2 and R replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if √ −1 ∈ K. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case K = R.2010 Mathematics Subject Classification. 05E40, 13C15, 05C62, 05E99.Proposition 2.3. The ideals in < (I Kn ) and in < (I K m,n−m ) have a linear resolution, height(in < (I Kn )) = n, height(in < (I K m,n−m )) = n − 1, depth(S/ in < (I Kn )) = 1 and S/ in < (I K m,n−m ) is Cohen-Macaulay. The same statements hold for the ideals I Kn and I K m,n−m .Proof. We first show that in < (I Kn ) has a linear resolution. By Lemma Lemma 2.1,We order the generators of this initial ideal in a way that the monomial generators of J 1 are bigger than the monomial generators of J 2 and such that the generators of J 1 as well as the generators of J 2 are ordered lexicographically induced by x 1 > · · · > x n > y 1 > · · · > y n . The ideal J has linear quotients with respect to this ordering of its monomial generators. Indeed, the ideal J 1 is known to have linear quotients. Now let x i y j ∈ J 2 . We denote by J ij the ideal generated by all monomial generators of J which are bigger than x i y j . Then J ij : x i y j = (x 1 , . . . , x n , y i+1 , . . . , y j−1 ). This shows that J has linear quotients and hence has a linear resolution by [4, Proposition 8.2.1]. Moreover it follows from [4, Corollary 8.2.2] that proj dim(J) = 2n − 2, because J 1n

Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

“…First observe that, for obvious reasons, L k G (1) is radical, it is a complete intersection if and only if G is a matching and it is prime if and only if G has no edges. For d = 2 the following result from [23] gives a complete answer for two of the three properties under discussion. (2) 1 2 3 4 5…”

Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

“…Recalling the definition of an OR from Definition 3.1 they ask: under what conditions is the set of ORs irreducible? Some progress on this question is reported in [23].…”

Binomial edge ideals of bipartite graphs

“…It is worth noting that, the main theorem gives also a classification of other classes of Cohen-Macaulay binomial ideals associated with bipartite graphs, Corollary 6.2: Lovász-Saks-Schrijver ideals [11], permanental edge ideals [11,Section 3] and parity binomial edge ideals [12].…”